ON MONOIDS AND DOMAINS WHOSE MONADIC SUBMONOIDS ARE KRULL
ANDREAS REINHART
Abstract. A submonoid S of a given monoid H is called monadic if it is a divisor-closed submonoid
of H generated by one element (i.e., there is some (non-zero) b ∈ H such that S is the smallest divisorclosed submonoid of H such that b ∈ S). In this paper we study monoids and domains whose monadic
submonoids are Krull monoids. These monoids resp. domains are called monadically Krull. Every
Krull monoid is a monadically Krull monoid, but the converse is not true. We provide several types of
counterexamples and present a few characterizations for monadically Krull monoids. Furthermore, we
show that rings of integer-valued polynomials over factorial domains are monadically Krull. Finally, we
investigate the connections between monadically Krull monoids and generalizations of SP-domains.
1. Introduction
The main goal of this paper is to study so called monadically Krull monoids (i.e. monoids where every
divisor-closed submonoid generated by one element is a Krull monoid). Studying monoids “monadically”
(i.e. investigating properties that are satisfied by all divisor-closed submonoids generated by one element)
is reasonable, since some types of monoids are better situated in the “local” than in the “global” situation.
On the other hand it turns out that there are a lot of monoid theoretical properties that are satisfied
by the monoid if and only if they are satisfied “monadically” (e.g., being atomic, completely integrally
closed, factorial). However, the Krull property does not behave like this (as pointed out in this work),
and thus monadically Krull monoids are of special interest. Being monadically Krull is related to “weak
factorization properties” that have been studied in a series of papers (see [8, 9, 23, 24, 25]). Moreover,
some recent work in studying monoids “monadically” has been done in [14, 22]. Investigating monadically
Krull monoids was also motivated by a problem that we want to discuss in more detail. Let R be a
(possibly noncommutative) ring and let C be a class of finitely generated (right) R-modules which is
closed under finite direct sums, direct summands, and isomorphisms. Then the set V(C) of isomorphism
classes of modules is a commutative semigroup with operation induced by the direct sum. This semigroup
encodes all possible information about direct sum decompositions of modules in C (see [5, 11]). If the
endomorphism ring of each module in C is semilocal, then V(C) is a Krull monoid ([10, Theorem 3.4]).
Moreover, every reduced Krull monoid can be realized by such a monoid of modules ([12]). Thus the
(global) property that V(C) is Krull follows from a family of local data, namely that all EndR (M ) are
semilocal. Furthermore, the assumption that EndR (M ) is semilocal implies that the smallest divisorclosed submonoid of V(C) generated by the class of M (denoted by add(M )) is a Krull monoid ([4, 5, 6]).
In the second section we will discuss the most important terminology. We give a brief introduction to
finitary ideal systems to simplify and unify the terminology about various types of ideals (e.g. ring ideals
and t-ideals).
In the third section we will prove that several interesting properties (like being completely integrally
closed, being atomic or being an FF-monoid) can be characterized by using the divisor-closed submonoids
generated by one element. Moreover, we provide another characterization of Krull monoids. The main
result in this section is a characterization of monadically Krull monoids. It turns out that the monadically
2000 Mathematics Subject Classification. 13A15, 13F05, 20M11, 20M12.
Key words and phrases. monadically, integer-valued, Krull monoid, Mori set, SP-domain.
This work was supported by the Austrian Science Fund FWF, Project Number P21576-N18.
1
2
ANDREAS REINHART
Krull monoids are precisely the atomic, completely integrally closed monoids where special sets of atoms
are finite up to associates.
In the fourth section we deal with the question whether every monadically Krull monoid is already a Krull
monoid. We provide several counterexamples. First we present a ring theoretical counterexample and
later we will introduce a counterexample in the monoid setting that is substantially stronger. The second
example will show that radical factorial FF-monoids (they are always monadically Krull) also need not
be Krull. By the way we answer some questions that have been raised in the literature in the negative.
In [9] it has been shown that every atomic IDPF-domain that contains a field of characteristics zero is
already completely integrally closed. We will point out that such a domain is not necessarily a Krull
domain. Furthermore, we deal with the problem whether the t-dimension of a t-SP-monoid (which is
some sort of generalized Krull monoid) is bounded by one and show that t-SP-monoids whose height-one
prime t-ideals are divisorial are not necessarily Krull. Moreover, it is well known that an integral domain
is a Prüfer domain that does not have non-zero idempotent prime ideals if and only if each of its primary
ideals is a power of its radical and its set of prime ideals satisfies the ACC (for example see [29, Corollary
5.5]). We show that the “t-analogue” of this statement is not true in the monoid setting.
In the fifth section we investigate rings of integer-valued polynomials. We prove that rings of integervalued polynomials over factorial domains are monadically Krull. Using this result we are able to provide
a large class of monadically Krull domains that are t-Prüfer domains and that fail to be Krull.
In the last section we deal with the question whether every radical factorial FF-domain of Krull dimension
one is already a Krull domain. Although we could not solve this problem so far, we will present partial
solutions. For example we will construct a BF-domain that is an SP-domain but not a Krull domain
(note that every radical factorial domain of Krull dimension one is an SP-domain, see [29, Proposition
3.11]). The counterexamples in this section are based on a construction used in [19]. Furthermore, we
investigate how far SP-domains (and their generalizations) are from being monadically Krull by studying
a special necessary property that pops up in the characterization of monadically Krull (in [23] this special
property is called pseudo-IDPF).
2. Preliminaries
In the following, a monoid is a commutative semigroup (multiplicatively written if not stated otherwise)
that possesses an identity and (if not stated otherwise) a zero element different from the identity such
that every non-zero element is cancellative. A quotient monoid of a monoid H is a monoid containing
H as a submonoid where every non-zero element is invertible and that is minimal with respect to this
property.
Let H be a monoid, K a quotient monoid of H and X ⊆ H. Set H • = H\{0}.
• For A, B ⊆ K let (A :K B) = {z ∈ K | zB ⊆ A}, A−1 = (H :K A) and Av = (A−1 )−1 .
• X is called (H-)divisor-closed if for all x ∈ H and y ∈ H • such that xy ∈ X it follows that x ∈ X.
• By [X]H (resp. [[X]]H ) we denote the smallest (divisor-closed) submonoid of H that contains X.
• If a ∈ H, set [[a]]H = [[{a}]]H .
• X is called an (H-)Mori set if for every F ⊆ X there exists some finite E ⊆ F such that E −1 = F −1 .
Note that [[a]]H = {b ∈ H | b |H an for some n ∈ N} ∪ {0} for all a ∈ H • .
Since we will use a slightly different version of ideal systems than those dealt with in [21], we will recall
the definition. The ideal systems in this work will always be ideal systems in the sense of [21] (but not
conversely). Let P(H) be the power set of H and r : P(H) → P(H) be a map. The map r is called a
(finitary) ideal system on H if the following properties are satisfied for all X, Y ⊆ H and c ∈ H.
• XH ∪ {0} ⊆ r(X).
• r(cX) = cr(X).
• If X ⊆ r(Y
S ), then r(X) ⊆ r(Y ).
• (r(X) = E⊆X,|E|<∞ r(E).)
ON MONOIDS AND DOMAINS WHOSE MONADIC SUBMONOIDS ARE KRULL
3
Note that if H • 6= H × , then v : P(H) → P(H)
S defined by v(X) = Xv for all X ⊆ H is an ideal system on
H and t : P(H) → P(H) defined by t(X) = E⊆X,|E|<∞ Ev for all X ⊆ H is a finitary ideal system on
H. If R is an integral domain, then d : P(R) → P(R) defined by d(X) = (X)R for all X ⊆ R is a finitary
ideal system on R. Furthermore, s : P(H) → P(H) defined by s(X) = XH if ∅ =
6 X ⊆ H and s(∅) = {0}
is a finitary ideal system on H.
In the following we will use most of the definitions and notations in [15] and [21] without further reference.
Especially, we will freely use the following terms: “BF-monoid”, “FF-monoid”, “ACCP”, “atomic”,
“factorial”, “Krull”, “completely integrally closed”, “valuation monoid”, “v-closed”, “root-closed” and
“GCD-monoid”. A monoid is called a Mori monoid if it is v-noetherian in the terminology of [15].
Observe that Mori sets defined in this work differ from those introduced in [28]. Note that if S ⊆ H
is a divisor-closed submonoid and H is a Krull monoid (a Mori monoid, a completely integrally closed
monoid), then S has the same property by [15, Proposition 2.4.4.2].
3. Monadic properties and Mori sets
First we present a simple characterization of being a Mori set. Using this result it is straightforward to
prove that H is a Mori monoid if and only if H is a Mori set.
Lemma 3.1. Let H be a monoid and X ⊆ H. Then X is not a Mori set if and only if there is some
(ai )i∈N ∈ X N such that {ai | i ∈ [1, n + 1]}−1 $ {ai | i ∈ [1, n]}−1 for all n ∈ N.
Proof. “⇒”: Let X be not a Mori set. Then there is some F ⊆ X such that for every finite E ⊆ F
it follows that F −1 $ E −1 . There exists some a1 ∈ F . Now let n ∈ N and (ai )ni=1 ∈ F [1,n] . Then
F −1 $ {ai | i ∈ [1, n]}−1 , and thus F * {ai | i ∈ [1, n]}v . Consequently, there exists some an+1 ∈ F \{ai |
i ∈ [1, n]}v , and thus {ai | i ∈ [1, n + 1]}−1 $ {ai | i ∈ [1, n]}−1 . Hence there is some (ai )i∈N ∈ F N
such that {ai | i ∈ [1, n + 1]}−1 $ {ai | i ∈ [1, n]}−1 for all n ∈ N. “⇐”: Let F = {ai | i ∈ N}.
Assume that there is some finite E ⊆ F such that E −1 = F −1 . Then there exists some n ∈ N such that
E ⊆ {ai | i ∈ [1, n]}. This implies that F −1 ⊆ {ai | i ∈ [1, n + 1]}−1 $ {ai | i ∈ [1, n]}−1 ⊆ E −1 = F −1 , a
contradiction.
Next we specify Krull monoids by using Mori sets. Note that the equivalence of 1 and 4 is well known.
Proposition 3.2. Let H be a monoid. The following conditions are equivalent:
1. H is a Krull monoid.
2. H is atomic, completely integrally closed and A(H) is a Mori set.
3. H is completely integrally closed and there is some Mori set F ⊆ H such that H = [F ∪ H × ]H .
4. H is completely integrally closed and every t-maximal t-ideal of H is divisorial.
Proof. 1. ⇒ 2.: Clear. 2. ⇒ 3.: Set F = A(H). Since H is atomic we have H = [F ∪ H × ]. 3. ⇒ 4.: Let
F ⊆ H be a Mori set such that H = [F ∪ H × ]H , P a t-maximal t-ideal of H and x ∈ P • . Then there
Q
(F )
are some ε ∈ H × and (αe )e∈F ∈ N0 such that x = ε e∈F eαe . Therefore, there exists some e ∈ F
such that e ∈ P and x ∈ eH. It follows that x ∈ {e}t ⊆ (P ∩ F )t . Consequently, P ⊆ (P ∩ F )t . There
is some finite E ⊆ P ∩ F such that E −1 = (P ∩ F )−1 . This implies that P ⊆ (P ∩ F )t ⊆ (P ∩ F )v =
Ev = Et ⊆ (P ∩ F )t ⊆ P , hence P = Ev , and thus P is divisorial. 4. ⇒ 1.: Let I be a non-zero
t-ideal of H. It is sufficient to show that I is t-invertible. Since H is completely integrally closed, it
follows that (II −1 )v = H. Assume that (II −1 )t $ H. Then there exists some t-maximal t-ideal P of H
such that (II −1 )t ⊆ P . We have H = (II −1 )v ⊆ ((II −1 )t )v ⊆ Pv = P , a contradiction. Consequently,
(II −1 )t = H.
Now we provide a few minor results about Mori sets and divisor-closed submonoids to prepare for the
main result in this section.
Lemma 3.3. Let H be a monoid, S ⊆ H a divisor-closed submonoid and X ⊆ S a subset. If X is an
H-Mori set, then X is an S-Mori set.
4
ANDREAS REINHART
Proof. Let K be a quotient monoid of H and L ⊆ K the quotient monoid of S. First we show that for
every Y ⊆ S it follows that (H :L Y ) = (S :L Y ). Let Y ⊆ S. “⊆”: Let x ∈ (H :L Y ), then xY ⊆ H.
Since S ⊆ H is divisor-closed it follows that xY ⊆ H ∩ L = S, hence x ∈ (S :L Y ). “⊇”: Trivial. Now let
X be an H-Mori set and F ⊆ X. Then there exists some finite E ⊆ F such that (H :K F ) = (H :K E).
This implies that (S :L F ) = (H :L F ) = (H :K F ) ∩ L = (H :K E) ∩ L = (H :L E) = (S :L E), hence X
is an S-Mori set.
Let H be a monoid, x ∈ H and n ∈ N and let A be some property that can be stated in the language of
monoids (e.g. atomic, Krull, Mori).
• Set Dn (x) = {u ∈ A(H) | u |H xn }.
• A submonoid S ⊆ H is called monadic if S = [[a]] for some a ∈ H • .
• We say that H is monadically A (or H is a monadically A monoid) if every monadic submonoid of
H satisfies A.
• The property A is said to be monadic (for H) if H has property A if and only if every monadic
submonoid of H has property A.
• If H is an integral domain we say that H is a A domain if H satisfies A as a monoid.
Note that H is monadically A if and only if [[E]] satisfies A for all non-empty finite E ⊆ H • .
Proposition 3.4. Let H be a monoid and K a quotient monoid of H.
S
1. H × = [[a]]× and A([[a]]) = A(H) ∩ [[a]] = n∈N Dn (a) for all a ∈ H • .
2. H is atomic if and only if H is monadically atomic.
3. H is completely integrally closed if and only if H is monadically completely integrally closed.
4. H is an FF-monoid if and only if H is a monadically FF-monoid.
Proof. 1. Let a ∈ H • . Clearly, [[a]]× ⊆ H × . If ε ∈ H × , then εε−1 = 1 ∈ [[a]], hence ε, ε−1 ∈ [[a]], and
thus ε ∈ [[a]]× . If x ∈ A([[a]]) and b, c ∈ H are such that x = bc, then b, c ∈ [[a]], hence b ∈ [[a]]× = H × or
×
×
c ∈ [[a]]× = H × . Finally, if x ∈ A(H) ∩ [[a]]
S and b, c ∈ [[a]] are suchn thatSx = bc, then b ∈ H = [[a]] or
×
×
c ∈ H = [[a]] . Obviously, A(H) ∩ [[a]] = n∈N {u ∈ A(H) | u |H a } = n∈N Dn (a).
2. This is an immediate consequence of 1.
3. “⇒”: Trivial “⇐”: Let x ∈ K • be almost integral over H. There exists some c ∈ H • such that
cxn ∈ H for all n ∈ N and there are some a, b ∈ H • such that x = ab . Let L ⊆ K be the quotient monoid
of [[abc]]. Then c ∈ [[abc]]• , x ∈ L and cxn ∈ H ∩ L = [[abc]] for all n ∈ N. Since [[abc]] is completely
integrally closed we have x ∈ [[abc]] ⊆ H.
4. “⇒”: This follows from 1 and [15, Theorem 1.5.6.2]. “⇐”: Let x ∈ H • . It is an easy consequence of 1
that f : {y[[x]] | y ∈ [[x]], y |[[x]] x} → {yH | y ∈ [[x]], y |[[x]] x} defined by f (I) = IH is a bijective map. Since
{y[[x]] | y ∈ [[x]], y |[[x]] x} is finite, we have {yH | y ∈ H, y |H x} = {yH | y ∈ [[x]], y |[[x]] x} is finite.
Let H be a monoid and K a quotient monoid of H.
• H is called seminormal if for all x ∈ K such that x2 , x3 ∈ H we have x ∈ H.
• H is called weakly factorial if every x ∈ H • \H × is a finite product of primary elements of H (i.e.
of elements x ∈ H • such that xH is primary).
• H is called radical factorial√if every x ∈ H • is a finite product of radical elements of H (i.e. of
elements x ∈ H • such that xH = xH).
We leave to the reader to prove that “satisfying the ACCP”, “seminormal”, “root-closed”, “atomic and
weakly factorial”, “atomic and radical factorial”, “being a BF-monoid”, “being a valuation monoid”,
“being a GCD-monoid” and “factorial” are also monadic properties for H. We do not know whether
“weakly factorial”, “radical factorial” and “v-closed” are monadic properties. If H is weakly factorial
(resp. radical factorial), then H is monadically weakly factorial (resp. monadically radical factorial) and
the following holds.
Remark 3.5. Let H be a monoid that is monadically v-closed. Then H is v-closed.
ON MONOIDS AND DOMAINS WHOSE MONADIC SUBMONOIDS ARE KRULL
5
Proof. Let K be a quotient monoid of H, ∅ 6= E ⊆ H • finite and x ∈ K • such that xE ⊆ Ev . There are
some y, z ∈ H • such that x = yz . Set S = [[E ∪ xE ∪ {y, z}]]. Observe that S is a monadic submonoid of
H, and thus S is vS -closed. Let L ⊆ K be the quotient monoid of S. Clearly, x ∈ L and it follows by
[15, Proposition 2.4.2.3] that xE ⊆ Ev ∩ S ⊆ (EvS )v ∩ S = EvS , hence x ∈ S. Consequently, x ∈ H. Now we present the main result in this section. It connects monadically Krull monoids with concepts
that are well known in the literature.
Theorem 3.6. Let H be a monoid. The following conditions are equivalent:
1. H is a monadically Krull monoid.
2. H is atomic and completely integrally closed and {uH | u ∈ A([[a]])} is finite for all a ∈ H • .
3. H is a completely integrally closed FF-monoid and for all a ∈ H • , A([[a]]) ⊆ Dk (a) for some k ∈ N.
4. H is atomic and completely integrally closed and A([[a]]) is an H-Mori set for all a ∈ H • .
Proof. 1. ⇒ 2.: By Propositions 3.4.2 and 3.4.3 we have H is atomic and completely integrally closed.
Let a ∈ H • . If P ∈ X([[a]]), then there is some b ∈ P • , hence there are some c ∈ H and n ∈ N such that
bc = an . It follows that c ∈ [[a]], hence an ∈ P , and thus a ∈ P . Therefore, a ∈ P for all P ∈ X([[a]]).
It follows by [15, Proposition 2.2.4.2] and [15, Theorem 2.2.5.2] that X([[a]]) is finite. It can be easily
deduced from [15, Theorem 2.7.14] that {u[[a]] | u ∈ A([[a]])} is finite. It follows by Proposition 3.4.1 that
f : {u[[a]] | u ∈ A([[a]])} → {uH | u ∈ A([[a]])} defined by f (I) = IH is bijective, hence {uH | u ∈ A([[a]])}
is finite.
2. ⇒ 3.: Let a ∈ H • . It follows that {uH | u ∈ D1 (a)} ⊆ {uH | u ∈ A([[a]])} is finite which implies
(together with the fact that H is atomic) that H is an FF-monoid. On the other hand there is some finite
E ⊆ A([[a]]) such that {uH | u ∈ A([[a]])} = {uH | u ∈ E}. There is some k ∈ N such that E ⊆ Dk (a).
Let u ∈ A([[a]]). Then some ε ∈ H × and some v ∈ E exist such that u = εv. Since v ∈ Dk (a), we
immediately obtain that u ∈ Dk (a).
3. ⇒ 4.: Let a ∈ H • and F ⊆ A([[a]]). There is some k ∈ N such that F ⊆ Dk (a) and since H is an
FF-monoid we have {uH | u ∈ F } ⊆ {uH | u ∈ Dk (a)} is finite. Consequently, there is some finite E ⊆ F
such that {uH | u ∈ F } = {uH | u ∈ E}. If x ∈ K, then x ∈ E −1 if and only if xuH ⊆ H for all u ∈ E
if and only if xuH ⊆ H for all u ∈ F if and only if x ∈ F −1 . Therefore, E −1 = F −1 .
4. ⇒ 1.: Let a ∈ H • . By Propositions 3.4.2 and 3.4.3 it follows that [[a]] is atomic and completely
integrally closed. Lemma 3.3 implies that A([[a]]) is an [[a]]-Mori set. By Proposition 3.2 we obtain that
[[a]] is a Krull monoid.
Using the terminology in [23] we obtain by Theorem 3.6 that H is a monadically Krull monoid if and only
if it is an atomic, completely integrally closed IDPF-monoid if and only if it is a completely integrally
closed FF-monoid that is a pseudo-IDPF monoid. We will see later that monadically Krull monoids are
not necessarily Krull monoids. Especially, we have that monadically Mori monoids are not necessarily
Mori monoids. The next result shows that if the Mori property is satisfied by a bigger class of divisorclosed submonoids, then the monoid itself satisfies the Mori property.
Proposition 3.7. Let H be a monoid. Then H is a Mori monoid if and only if [[X]] is a Mori monoid
for every denumerable subset X ⊆ H • .
Proof. Let K be a quotient monoid of H. “⇒”: Trivial. “⇐”: Assume that H is not a Mori monoid.
Then H is not a Mori set. By Lemma 3.1 there exists some (ai )i∈N ∈ H N such that (H :K {ai | i ∈
[1, n + 1]}) $ (H :K {ai | i ∈ [1, n]}) for all n ∈ N. Therefore, there exist some (xi )i∈N ∈ H N and
(yi )i∈N ∈ (H • )N such that for all n ∈ N we have an+1 xynn ∈ K\H and ai xynn ∈ H for all i ∈ [1, n].
Let S = [[{an xn yn | n ∈ N}]] and let L ⊆ K be the quotient monoid of S. Then (ai )i∈N ∈ S N and
( xyii )i∈N ∈ LN . Moreover, we have for all n ∈ N that an+1 xynn ∈ L\S and ai xynn ∈ H ∩ L = S for all
i ∈ [1, n]. This implies that (S :L {ai | i ∈ [1, n + 1]}) $ (S :L {ai | i ∈ [1, n]}) for all n ∈ N. It follows by
Lemma 3.1 that S is not an S-Mori set, and thus S is not a Mori monoid, a contradiction.
6
ANDREAS REINHART
4. Counterexamples
It is of interest to know whether every monadically Krull monoid is already a Krull monoid. In this
section we prove that this is not necessarily true and show that even strong improvements of monadically
Krull can fail to be Krull. For technical reasons we will consider (multiplicatively written) monoids that
do not posses a zero element in this section. Moreover, we will use monoids that are additively written
(the zero element is their identity and they do not posses an “additive” analogue of a “multiplicative”
zero element). All terminology that has been introduced so far can be adapted in an obvious way for
these types of monoids. Observe that the label “quotient monoid” will be replaced by “quotient group”
for monoids without a zero element. Note that a monoid is root closed if and only if it is integrally closed
in terms of [16]. We want to thank F. Kainrath who led our attention to the integral domain constructed
in the next example.
Example 4.1. There exists a monadically Krull domain that is not a Krull domain.
Proof. Let R be an integrally closed noetherian domain, (Xi )i∈N0 a sequence of independent
indetermiQn
nates over R and K a field of quotients of R[{Xi | i ∈ N0 }]. For n ∈ N0 set Sn = R[{ i=0 Xiai | (ai )ni=0 ∈
Pn
S
[0,n]
N0 , a0 ≥ i=1 a2ii }]. Let S = n∈N Sn . We show that S is a monadically Krull domain that is not a
Krull domain. Note that S is a subring of R[{Xi | i ∈ N0 }] and K is a field of quotients of S.
Qn
i
2n+1 −1
First we show that S is not a Krull domain. For n ∈ N0 set an = X0n+1 ( i=1 Xi2 )Xn+1
. By Lemma
3.1 it is sufficient to show that Xk+1 ∈ {ai | i ∈ [0, k]}−1 \{ai | i ∈ [0, k + 1]}−1 for all k ∈ N0 . Let
Pi
i+1
j
1
1
1
= i + 1 − 2i+1
≤ i + 1 it follows that
+ 2k+1
k ∈ N0 and i ∈ [0, k]. Since j=1 22j + 2 2i+1−1 + 2k+1
Q
Pk
i+1
j
j
i
i+1
2
−1
2
Xk+1 ai = X0 ( j=1 Xj )Xi+1 Xk+1 ∈ S, and thus Xk+1 ∈ {aj | j ∈ [0, k]}−1 . Since j=1 22j +
Qk
k+2
j
2k+1 +1 2k+2 −1
1
2k+1 +1
+ 2 2k+2−1 = k + 2 + 2k+2
> k + 2, we have Xk+1 ak+1 = X0k+2 ( j=1 Xj2 )Xk+1
Xk+2
6∈ S,
2k+1
hence Xk+1 6∈ {aj | j ∈ [0, k + 1]}−1 .
Next we prove that Sl is a divisor-closed subring of S that is noetherian and integrally closed for all
l ∈ N. Let l ∈ N. Clearly, Sl is a subring of S. Let f, g ∈ S • be such that f g ∈ Sl . There is some m ∈ N
such that m ≥ l and f, g ∈ Sm . Since f g ∈ R[{Xi | i ∈ [0, l]}], it follows that f ∈ R[{Xi | i ∈ [0, l]}],
hence f ∈ Sm ∩ R[{Xi | i ∈ [0, l]}] = Sl . Therefore, Sl is a divisor-closed subring of S.
Ql
Ql
Pl
Pl
[0,l]
[0,l]
Set T = { i=0 Xiai | (ai )li=0 ∈ N0 , a0 ≥ i=1 a2ii }, U = { i=0 Xiai | (ai )li=0 ∈ N0 , l ≥ a0 ≥ i=1 a2ii }
and V = [U ]. Then T is a submonoid of Sl . We show that T is root closed and T = V . Let L ⊆ K be
Ql
[0,l]
a quotient group of T . Let x ∈ L and r ∈ N be such that xr ∈ T . Since { i=0 Xiai | (ai )li=0 ∈ N0 }
[0,l]
is a root closed submonoid of L that contains T it follows that there is some (bi )li=0 ∈ N0 such that
Pl bi
Ql
Ql
Pl
i
x = i=0 Xibi . This implies that i=0 Xirbi ∈ T , and thus rb0 ≥ i=1 rb
i=1 2i ,
2i . Consequently, b0 ≥
hence x ∈ T . Therefore, T is root closed. It remains to prove by induction on k that for all k ∈ N and
Pl
Ql
[1,l]
all (ai )li=1 ∈ N0 such that k ≥ i=1 a2ii it follows that X0k i=1 Xiai ∈ V . If k = 1, then the assertion
Pl ai
[1,l]
is clear. Now let k ∈ N and (ai )li=1 ∈ N0 be such that k + 1 ≥
i=1 2i . Case 1: There is some
Pl
aj −2j
ai
j
j ∈ [1, l] such that aj > 2 : We have k ≥ i=1,i6=j 2i + 2j . It follows by the induction hypothesis that
Ql
Ql
Ql
j
a −2j
a −2j
X0k ( i=1,i6=j Xiai )Xj j
∈ V . Therefore, X0k+1 i=1 Xiai = X0 Xj2 X0k ( i=1,i6=j Xiai )Xj j
∈ V . Case
Pl ai
ai
k+1 Ql
j
2: aj ≤ 2 for all j ∈ [1, l]: We have i=1 2i ≤ l. If l ≥ k + 1, then X0
i=1 Xi ∈ V , by definition.
ai
k+1 Ql
k+1−l l Ql
Now let l < k + 1. Since X0 ∈ V , it follows that X0
X0 i=1 Xiai ∈ V . By [17,
i=1 Xi = X0
Corollary 15.12] it follows that Sl is noetherian and integrally closed.
Let a ∈ S • . There is some s ∈ N such that a ∈ Ss . Since Ss is a divisor-closed subring of S it follows
that [[a]]S is a divisor-closed submonoid of Ss . Since Ss is a Krull domain this implies that [[a]]S is a Krull
monoid. Consequently, S is a monadically Krull domain.
It has been pointed out in [9] that every atomic IDPF-domain (this notion has been introduced in [23])
that contains a field of characteristics zero is already completely integrally closed. Now let the domain
ON MONOIDS AND DOMAINS WHOSE MONADIC SUBMONOIDS ARE KRULL
7
R in the last example be a field of characteristics zero. Then the domain S in the last example is an
atomic IDPF-domain that contains a field of characteristics zero and yet S is not a Krull domain. Let
H be a monoid and r a finitary ideal system on H. Let J ⊆ P(H). We say that J possesses a length
function if there exists some map λ : J → N0 such that λ(J) < λ(I) for all I, J ∈ J such that I $ J.
Note if J possesses a length function, then J satisfies the ACC. Moreover, if R is an integral domain,
then the set of non-zero ideals of R possesses a length function if and only if R is a noetherian domain
and dim(R) ≤ 1. Observe that a monoid H is a BF-monoid if and only if {xH | x ∈ H • } possesses a
length function. Moreover, possessing a length function is in some sense the same as satisfying a strong
version of the ACC. We use it in the next example to highlight that it is not only a BF-monoid but also
that the set of radicals of principal ideals possesses a length function. Next we introduce several other
types of generalizations of the Krull property to study the following example in detail.
• H is called an r-SP-monoid if every r-ideal of H is a finite r-product of radical r-ideals of H.
• H is called an r-Prüfer monoid if every non-zero r-finitely generated r-ideal of H is r-invertible.
• An r-ideal I of H is called r-cancellative if I is cancellative in the r-ideal semigroup of H.
Clearly, H is a Krull monoid if and only if it is a Mori monoid that is a t-Prüfer monoid. Note that if H
is a radical factorial monoid, then for all a ∈ H • we have A([[a]]) ⊆ D1 (a). Using Theorem 3.6 and [29,
Proposition 2.4] it is straightforward to prove that every radical factorial FF-monoid is a monadically
Krull monoid (but even a Krull monoid needs not be radical factorial, see [29, Example 4.3]). In this light
we will sharpen our first counterexample in the monoid setting and prove that even radical factorial FFmonoids can fail to be Krull. A sequence (xi )i∈N0 of integers is called formally infinite if {i ∈ N0 | xi 6= 0}
Pk
is finite. If H is additively written, k ∈ N and I ⊆ H, then set kI = { i=1 ai | (ai )ki=1 ∈ I [1,k] }.
(N0 )
Example 4.2. Let G be a free abelian group with basis
denote
P (ei )i∈N0 . For x ∈ G, let (xi )i∈N0 ∈ Z
the unique formally infinite sequence such that x = i∈N0 xi ei . Set H = {x ∈ G | x0 ≥ xi ≥ 0 for all
i ∈ N0 }. Then H is a submonoid of G, G is a quotient group of H and the following is true:
1. v-spec(H)• = X(H), t-spec(H)• = X(H) ∪ {H\H × }, every non-empty t-ideal of H is t-cancellative
and (kP )t is P -primary for all k ∈ N and P ∈ t-spec(H)• . √
2. H is a t-SP-monoid, t-dim(H) = 2, H is an FF-monoid, { y + H | y ∈ H} possesses a length
function and every radical element of H is either an atom or a unit.
In particular, H is a radical factorial monoid that is neither a Mori monoid nor a t-Prüfer monoid.
Proof. Clearly, H is a submonoid of G and H × = {0}. Let K ⊆ G be the quotient group of H and
i ∈ N0 . Obviously, e0 , e0 + ei ∈ H, hence ei = e0 + ei − e0 ∈ K. Therefore, G = K is a quotient
0
group of H. For r ∈ NN
0 set Ir = {x ∈ G | x0 ≥ xj+1 + r2j+1 , xj ≥ r2j for all j ∈ N0 }. Set
N0
0
I = {r ∈ N0 | |{j ∈ N0 | r2j 6= 0}| < ∞, r0 ≥ r2j+1 + r2j+2 for all j ∈ N0 } and L = {r ∈ NN
0 | |{j ∈
(i)
N
N0 | r2j 6= 0}| < ∞, r0 = max{r2j+1 + r2j+2 | j ∈ N0 }}. For i ∈ N0 , let s(i) ∈ N0 0 be defined by sj = 1
(i)
0
if j ∈ {0, i} and sj = 0 if j ∈ N0 \{0, i}. If r, s ∈ NN
0 and k ∈ N0 , then we set r + s = (ri + si )i∈N0 ,
kr = (kri )i∈N0 and r ≤ s if rj ≤ sj for all j ∈√N0 .
yi > 0} ⊆ {i ∈ N0 | xi > 0}
Claim 1: For all x, y ∈ H it follows that x ∈ y + H if and only if {i ∈ N0 | √
and {i ∈ N0 | y0 > yi } ⊆ {i ∈ N0 | x0 > xi }. Let x, y ∈ H. Observe that x ∈ y + H if and only if there
is some k ∈ N such that kxi ≥ yi and k(x0 − xi ) ≥ y0 − yi for all i ∈ N0 . “⇒”: Let k ∈ N be such that
kxj ≥ yj and k(x0 −xj ) ≥ y0 −yj for all j ∈ N0 . Let i ∈ N0 . If yi > 0, then kxi ≥ yi > 0, and thus xi > 0.
If y0 > yi , then k(x0 − xi ) ≥ y0 − yi > 0, hence x0 > xi . “⇐”: Let {i ∈ N0 | yi > 0} ⊆ {i ∈ N0 | xi > 0}
and {i ∈ N0 | y0 > yi } ⊆ {i ∈ N0 | x0 > xi }. Set k = 1 + max{yi | i ∈ N0 }. Then k ∈ N and it is clear
that kxi ≥ yi and k(x0 − xi ) ≥ y0 − yi for all i ∈ N0 .
Claim 2: For all ∅ =
6 A ⊆ H we have Av = {x ∈ G | x0 ≥ xi + min{a0 − ai | a ∈ A} and xi ≥ min{ai |
a ∈ A} for all i ∈ N}. Let ∅ =
6 A ⊆ H. Set m(i) = min{a0 − ai | a ∈ A} and n(i) = min{ai | a ∈ A} for
−1
all i ∈ N. Observe that A = {x ∈ G | x + a ∈ H for all a ∈ A} = {x ∈ G | for all a ∈ A and i ∈ N,
x0 + a0 ≥ xi + ai ≥ 0} = {x ∈ G | x0 + m(i) ≥ xi and xi + n(i) ≥ 0 for all i ∈ N}. “⊆”: Let x ∈ Av and
8
ANDREAS REINHART
i ∈ N. Set y = m(i) ei and z = −n(i) ei . We have y, z ∈ A−1 , hence x + y ∈ H and x + z ∈ H. Therefore,
x0 + y0 ≥ xi + yi and xi + zi ≥ 0. This implies that x0 ≥ xi + m(i) and xi ≥ n(i) . “⊇”: Let x ∈ G be
such that x0 ≥ xj + m(j) and xj ≥ n(j) for all j ∈ N. Let y ∈ A−1 and i ∈ N. Then x0 ≥ xi + m(i) ,
y0 + m(i) ≥ yi , xi ≥ n(i) and yi + n(i) ≥ 0, hence x0 + y0 + m(i) ≥ xi + yi + m(i) and xi + yi + n(i) ≥ n(i) .
Therefore, x0 + y0 ≥ xi + yi ≥ 0. This implies that x + y ∈ H. Consequently, x ∈ Av .
As usual we denote by It (H)• (resp. Iv (H)• ) the set of non-empty t-ideals of H (resp. the set of nonempty divisorial ideals of H).
Claim 3: It (H)• = {Ir | r ∈ I} and Iv (H)• = {Ir | r ∈ L}. First we prove that It (H)• = {Ir | r ∈ I}.
“⊆”: Let I ∈ It (H)• . For i ∈ N0 set r2i = min{yi | y ∈ I} and r2i+1 = min{y0 − yi+1 | y ∈ I}. There
(2i)
(2i+1)
(2i+1)
is some sequence (z (j) )j∈N0 ∈ I N0 such that zi = r2i and z0
− zi+1 = r2i+1 for all i ∈ N0 . If
(0)
j ∈ N0 , then since z0
(0)
(0)
(0)
− zj+1 ≥ r2j+1 and zj+1 ≥ r2j+2 we obtain that r0 = z0
(0)
Moreover, |{j ∈ N0 | r2j 6= 0}| ≤ |{j ∈ N0 | zj
≥ r2j+1 + r2j+2 .
6= 0}| < ∞. Therefore, r ∈ I. It remains to show
(0)
that I = Ir . “⊆”: Trivial. “⊇”: Let x ∈ Ir . Set E = {i ∈ N | xi 6= 0 or zi 6= 0}. Then E is finite.
(2i−1)
(2i−1) (2i)
(2i)
(0)
(0)
It is sufficient to prove that x0 − xj ≥ min({z0
− zj
, z0 − zj | i ∈ E} ∪ {z0 − zj }) and
(2i−1)
xj ≥ min({zj
(2i)
, zj
(0)
| i ∈ E} ∪ {zj }) for all j ∈ N, because then x ∈ ({z (2i−1) , z (2i) | i ∈ E} ∪ {z (0) })v
(2j−1)
by claim 2, hence x ∈ I. Let j ∈ N. Case 1a: xj 6= 0. It follows that x0 − xj ≥ r2j−1 = z0
Case 1b: xj = 0. We have x0 − xj = x0 ≥ r0 =
(0)
z0
(2j)
≥
(0)
z0
−
(0)
zj .
(2j−1)
− zj
.
Case 2a: j ∈ E. It follows that
(0)
xj ≥ r2j = zj . Case 2b: j 6∈ E. We have xj = 0 = zj . “⊇”: Let r ∈ I and x ∈ (Ir )t . Then there is
some finite ∅ =
6 A ⊆ Ir such that x ∈ Av . It is an immediate consequence of claim 2 that x0 ≥ xj + r2j−1
and xj ≥ r2j for all j ∈ N. Since A is finite, there is some l ∈ N such that xl = 0 and al = 0 for all
a ∈ A. It follows by claim 2 that x0 ≥ xl + min{a0 − al | a ∈ A} = min{a0 | a ∈ P
A} ≥ r0 . Consequently,
x0 ≥ xj+1 + r2j+1 and xj ≥ r2j for all j ∈ N0 , and thus x ∈ Ir . Observe that i∈N0 r2i ei ∈ Ir , hence
Ir ∈ It (H)• .
Next we show that Iv (H)• = {Ir | r ∈ L}. “⊆”: Let I ∈ Iv (H)• . As in the preceding part of the proof
there is some r ∈ I such that I = Ir , r2i = min{yi | y ∈ I} and r2i+1 = min{y0 − yi+1 | y ∈ I} for all
i ∈ N0 . SetPs = max{r2i+1 + r2i+2 | i ∈ N0 }. It remains to show that s ≥ r0 , because then r ∈ L. Set
x = se0 + i∈N r2i ei . If i ∈ N, then x0 = s ≥ r2i−1 + r2i = xi + r2i−1 and xi = r2i . Therefore, claim 2
implies that x ∈ (Ir )v = Ir , and thus s = x0 ≥ r0 . “⊇”: Let r ∈ L and x ∈ (Ir )v . It follows by claim 2
that x0 ≥ xi + r2i−1 and xi ≥ r2i for all i ∈ N. There is some j ∈ N such that r0 = r2j−1 + r2j , hence
x0 ≥ xj + r2j−1 ≥ r2j + r2j−1 = r0 . This implies that x ∈ Ir .
Claim 4:
Ia+b = (Ia + Ib )t and Ia ⊆ Ib if andPonly if b ≤ a. Let a, b ∈ I. Set
P For all a, b ∈ I, P
y (0) = i∈N0 a2i ei , z (0) = i∈N0 b2i ei and for j ∈ N set y (j) = i∈N0 ,i6=j a2i ei + (a0 − a2j−1 )ej and
P
z (j) = i∈N0 ,i6=j b2i ei + (b0 − b2j−1 )ej . Observe that y (j) ∈ Ia and z (j) ∈ Ib for all j ∈ N0 . “⊆”: Let
(l)
(l)
(l)
(l)
x ∈ Ia+b . We prove that x0 ≥ xj + min{y0 + z0 − yj − zj
(l)
(l)
min{yj + zj
| l = 0 or l ∈ N, xl 6= 0} and xj ≥
| l = 0 or l ∈ N, xl 6= 0} for all j ∈ N, because then x ∈ {y (l) + z (l) | l = 0 or l ∈ N, xl 6= 0}v
(0)
by claim 2, hence x ∈ (Ia + Ib )t . Let j ∈ N. Clearly, xj ≥ a2j + b2j = yj
(j)
(j)
(j)
(j)
have xj + y0 + z0 − yj − zj
(0)
+ zj . Case 1: xj 6= 0. We
= xj + a0 + b0 − (a0 − a2j−1 ) − (b0 − b2j−1 ) = xj + a2j−1 + b2j−1 ≤ x0 .
(0)
(0)
(0)
(0)
Case 2: xj = 0. It follows that xj + y0 + z0 − yj − zj = a0 + b0 − a2j − b2j ≤ a0 + b0 ≤ x0 . “⊇”:
Obviously, Ia + Ib ⊆ Ia+b and a + b ∈ I. Therefore, claim 3 implies that (Ia + Ib )t ⊆ Ia+b .
(i)
Clearly, if b ≤ a, then Ia ⊆ Ib . Now let Ia ⊆ Ib . Note that y (i) ∈ Ib for all i ∈ N0 , hence y0 ≥
(i)
(i)
(j+1)
(j+1)
(0)
yj+1 + b2j+1 and yj ≥ b2j for all i, j ∈ N0 . If j ∈ N0 , then y0
≥ yj+1 + b2j+1 and yj ≥ b2j , hence
a0 ≥ a0 − a2j+1 + b2j+1 and a2j ≥ b2j . Consequently, ai ≥ bi for all i ∈ N0 , and thus b ≤ a.
Claim 5: t-spec(H)• = {Is(i) | i ∈ N0 } and X(H) = {Is(i) | i ∈ N}. First we show that t-spec(H)• =
{Is(i) | i ∈ N0 }. “⊆”: Let P ∈ t-spec(H)• . By claim 3 there is some r ∈ I such that P = Ir . Case 1:
rj = 0 for all j ∈ N. Since P 6= H, we have r0 6= 0. This implies that r = ks(0) for some k ∈ N, and thus
ON MONOIDS AND DOMAINS WHOSE MONADIC SUBMONOIDS ARE KRULL
9
0
P = (kIs(0) )t by claim 4. Therefore, P = Is(0) . Case 2: rj 6= 0 for some j ∈ N. Let a ∈ NN
0 be defined
by ai = ri if i ∈ N0 , i 6= j and ai = ri − 1 otherwise. Then a ∈ I, r ≤ s(j) + a and r a. Therefore,
claim 4 implies that (Is(j) + Ia )t ⊆ P and Ia * P , hence Is(j) ⊆ P . Note that s(j) ≤ r, and thus P = Is(j)
by claim 4. “⊇”: Observe that Is(2i) = {x ∈ H | xi ≥ 1} and Is(2i+1) = {x ∈ H | x0 ≥ xi+1 + 1} for all
i ∈ N0 . Using this and claim 3 it is straightforward to prove that Is(i) ∈ t-spec(H)• for all i ∈ N0 .
Next we show that X(H) = {Is(i) | i ∈ N}. “⊆”: Let P ∈ X(H). Then P ∈ t-spec(H)• , hence P = Is(i)
for some i ∈ N0 . Since s(0) < s(1) , it follows by claim 4 that Is(1) $ Is(0) , and thus i ∈ N. “⊇”: Let i ∈ N
and P ∈ t-spec(H)• be such that P ⊆ Is(i) . There is some j ∈ N0 such that P = Is(j) . It follows by claim
4 that s(i) ≤ s(j) , hence s(i) = s(j) . Therefore, P = Is(i) , and thus Is(i) ∈ X(H).
Claim 6: Ir is a radical t-ideal of H for all r ∈ I such that r0 = 1. Let r ∈ I be such that r0 = 1. By claim
√
T
T
(i)
4 and claim 5 we have Ir = P ∈t-spec(H)• ,P ⊇Ir P = i∈N0 ,s(i) ≤r Is(i) = {x ∈ G | x0 ≥ xj+1 +max{s2j+1 |
(i)
i ∈ N0 , s(i) ≤ r}, xj ≥ max{s2j | i ∈ N0 , s(i) ≤ r} for all j ∈ N0 } = {x ∈ G | x0 ≥ xj+1 + r2j+1 , xj ≥ r2j
for all j ∈ N0 } = Ir .
1. By claim 3 and claim 5 we have v-spec(H)• = t-spec(H)• ∩ Iv (H)• = {Is(i) | i ∈ N0 } ∩ {Ir | r ∈
L} = {Is(i) | i ∈ N} = X(H) and t-spec(H)• = X(H) ∪ {Is(0) } = X(H) ∪ {H\H × }. Let A, B, C ∈ It (H)•
be such that (A + B)t = (A + C)t . By claim 3 there exist some a, b, c ∈ I such that A = Ia , B = Ib
and C = Ic . It follows by claim 4 that Ia+b = (A + B)t = (A + C)t = Ia+c , and thus a + b = a + c by
claim 4. Consequently, b = c, hence B = C. Now let k ∈ N and P ∈ t-spec(H)• . By claim 4 we have
(kIs(2i) )t = Iks(2i) = {x ∈ H | xi ≥ k} and (kIs(2i+1) )t = Iks(2i+1) = {x ∈ H | x0 ≥ xi+1 + k} for all
i ∈ N0 . Using this it is straightforward to prove that (kP )t is P -primary.
2. It follows by 1 that t-dim(H) = 2, since H\H × is not divisorial. Let I ∈ It (H)• . By claim 3 there is
(i)
some r ∈ I such that I = Ir . For i ∈ [1, r0 ] and j ∈ N0 set rj = 1, if ((j is even and rj ≥ i) or (j is odd
(i)
0
and rj+1 < i ≤ rj + rj+1 )) and set rj = 0 otherwise. Observe that (r(i) )ri=1
∈ {a ∈ I | a0 = 1}[1,r0 ] and
Pr0 (i)
Pr0
r = i=1 r . By claim 4 we have I = ( i=1 Ir(i) )t . Moreover, Ir(i) is a radical t-ideal for all i ∈ [1, r0 ]
by claim 6. Therefore, H is a t-SP-monoid. Set F = {x ∈ G | xi ≥ 0 for all i ∈ N0 }. Obviously, F is a
free abelian √
monoid and H ⊆ F is a submonoid. Consequently,
√ H is an FF-monoid.
Set M = { y + H | y ∈ H} and let I ∈ M. Then I = x + H for some x ∈ H. Set m = |{i ∈
N0 | xi > 0}| and l = (m + 1)2 . Let K ⊆ M be a chain such that min(K) = I (where min(K) denotes
(L)
K
the smallest
√ element of K with respect to inclusion). There is some sequence (x )L∈K ∈ H such
that J = x(J) + H for all J ∈ K. Let f : K → [0, m] × [0, m] be defined by f (J) = (|{i ∈ N0 |
(J)
(J)
(J)
xi > 0}|, |{i ∈ N0 | x0 > xi > 0}|). Using claim 1 and the fact that min(K) = I it is easy to
prove that f is well-defined. We show that f is injective. Let J, L ∈ K be such that f (J) = f (L).
(L)
(J)
Without restriction let J ⊆ L. By claim 1 we have {i ∈ N0 | xi > 0} ⊆ {i ∈ N0 | xi > 0} and
(L)
(L)
(J)
(J)
{i ∈ N0 | x0 > xi > 0} ⊆ {i ∈ N0 | x0 > xi > 0}. Since f (J) = f (L), this implies that
(J)
(L)
(J)
(J)
(L)
(L)
{i ∈ N0 | xi > 0} = {i ∈ N0 | xi > 0} and {i ∈ N0 | x0 > xi > 0} = {i ∈ N0 | x0 > xi > 0}.
√
√
(J)
(J)
(L)
(L)
Consequently, {i ∈ N0 | x0 > xi } = {i ∈ N0 | x0 > xi }, and thus J = x(J) + H = x(L) + H =
L by claim 1. Since f is injective we have |K| ≤ l. Let λ : M → N0 be defined by λ(J) = max{|K| | K ⊆ M
is a chain and min(K) = J}. Using the previous it is easy to prove that λ is a well-defined map and
λ(J) < λ(L) for all J, L ∈ M such that L $ J. Consequently, M possesses a length function.
P
Now let y be a radical element of H. There is some k ∈ N such that yk = 0. Set x = 2e0 +ek + i∈N,yi >0 ei .
√
It follows by claim 1 that x ∈ y + H = y + H, hence x0 − y0 ≥ xi − yi ≥ 0 for all i ∈ N0 . Therefore,
2 − y0 = x0 − y0 ≥ xk − yk = 1, and thus y0 ≤ 1. Consequently, y ∈ A(H) ∪ H × .
Since H\H × is a t-ideal it follows that Ct (H) is trivial, and thus we have H is radical factorial by [29,
Proposition 3.10.2]. Moreover, since t-dim(H) = 2 we have H is not a Krull monoid. Therefore, H is not
a Mori monoid by [29, Proposition 2.6]. It follows by [29, Proposition 3.9] and [29, Corollary 3.14] that
H is not a t-Prüfer monoid.
10
ANDREAS REINHART
Note that if H is a discrete valuation monoid (i.e. an atomic valuation monoid H with H • 6= H × ), then
every radical element of H is either an atom or a unit. The last example also shares this property with
discrete valuation monoids. An integral domain is called an almost Krull domain if all its localizations
at prime ideals are Krull domains. The following question has been raised by Pirtle (see [27]): Is every
almost Krull domain whose height-one prime ideals are divisorial already a Krull domain? Arnold and
Matsuda answered Pirtle’s question in the negative (see [3]). Note that our last example is of similar
type, since it shows that a (radical factorial) t-SP-monoid whose height-one prime t-ideals are divisorial
is not necessarily a Krull monoid. This also answers the questions raised after Proposition 2.6 in [29] in
the negative. Finally, Example 4.2 shows that being a t-Prüfer monoid is not a monadic property and
being “primary r-ideal inclusive” in [29, Corollary 5.3 and Theorem 5.4] cannot be omitted.
5. Connections with rings of integer-valued polynomials
In this section we investigate the connections between rings of integer-valued polynomials and monadically
Krull monoids. In particular, we continue our search for examples of monadically Krull domains that
are not Krull. As in section four, we will consider additively written monoids that do not posses an
“additive” analogue of a “multiplicative” zero element.
Let R be an integral domain, K a field of quotients of R and X an indeterminate over K. If a, b ∈ R,
then we write a 'R b if b = ac for some c ∈ R× . Set Int(R) = {f ∈ K[X] | f (c) ∈ R for all c ∈ R}, called
the ring of integer-valued polynomials over R. Observe that R ⊆ R[X] and R ⊆ Int(R) are divisor-closed,
Int(R)× = R[X]× = R× and A(Int(R)) ∩ R = A(R[X]) ∩ R = A(R). Especially, if R[X] is monadically
Krull or Int(R) is monadically Krull, then R is monadically Krull.
Now let R be factorial and Q a system of representatives of prime elements of R. Recall that R[X]
is factorial. If T ⊆ R, then let GCDR (T ) be the set of all greatest common divisors of T (in R).
If g ∈ R[X]\R, then g is called primitive if GCDR[X] (g, c) = R[X]× for all c ∈ R• (equivalently:
Pk
GCDR ({ai | i ∈ [0, k]}) = R× for all k ∈ N0 and (ai )ki=0 ∈ R[0,k] such that g = i=0 ai X i ). If q ∈ Q,
then let vq : R → N0 ∪ {∞} denote the q-adic valuation of R. Let dQ : Int(R)• → R• be defined by
Q
dQ (g) = p∈Q pmin{vp (g(c))|c∈R} for all g ∈ Int(R)• . Set d = dQ . Note that d(g) ∈ GCDR ({g(c) | c ∈ R})
g
∈ Int(R) for all g ∈ Int(R)• .
and d(g)
If M is a set and l ∈ N, then a finite sequence (ai )li=1 ∈ M l , will be denoted by a (i.e. a = (ai )li=1 ).
Let n ∈ N, f ∈ (Int(R)• )n and x ∈ Nn0 \{0}. Then x is called f -irreducible if for all y, z ∈ Nn0 such
Qn
Qn
Qn
that x = y + z and d( i=1 fixi ) = d( i=1 fiyi )d( i=1 fizi ) it follows that y = 0 or z = 0 (this definition
does not depend on the choice of Q). In the next Lemma we will use [15, Definition 1.5.2] and Dickson’s
Theorem (see [15, Theorem 1.5.3]) without further citation.
Lemma 5.1. Let R be a factorial domain, n ∈ N and f ∈ (Int(R)• )n . Then {x ∈ Nn0 | x is f -irreducible}
is finite.
Qn
Proof. Let Q be a system of representatives of prime elements of R and T = {w ∈ R | ( i=1 fi )(w) 6= 0}.
We prove that min{vq (g(w)) | w ∈ R} = min{vq (g(w)) | w ∈ T } for all q ∈ Q and g ∈ Int(R)• . Let
q ∈ Q and g ∈ Int(R)• . Then min{vq (g(w)) | w ∈ R} = vq (g(v)) for some v ∈ R. Observe that
there is some k ∈ N such that vq (g(v)) = vq (g(v + q l )) for all l ∈ N≥k . Since R\T is finite, there is
some m ∈ N≥k such that v + q m ∈ T . We have min{vq (g(w)) | w ∈ R} = vq (g(v + q m )), and thus
min{vq (g(w)) | w ∈ R} = min{v
Qn q (g(w)) | w ∈ T }.
Set P = {p ∈ Q | min{vp (( i=1 fi )(w)) | w ∈ R} > 0}. Clearly, P is finite. If p ∈ P , then thereSis some
finite Sp ⊆ T such that Min({(vp (fi (w)))ni=1 | w ∈ T }) = {(vp (fi (w)))ni=1 | w ∈ Sp }. Set S = p∈P Sp .
Pn
Then S is finite. For γ ∈ S P set Ωγ = {u ∈ Nn0 | i=1 (vp (fi (w)) − vp (fi (γ(p))))ui ≥ 0 for all p ∈ P and
w ∈ S}. If γ ∈ S P , then Ωγ is an additive monoid and by [15, Theorem 2.7.14] and
S [15, Proposition 1.1.7.2]
we have A(Ωγ ) is finite. It suffices to show that {x ∈ Nn0 | x is f -irreducible} ⊆ γ∈S P A(Ωγ ). Let x ∈ Nn0
Pn
Pn
be f -irreducible. There is some δ ∈ S P such that min{ i=1 vp (fi (w))xi | w ∈ S} = i=1 vp (fi (δ(p)))xi
ON MONOIDS AND DOMAINS WHOSE MONADIC SUBMONOIDS ARE KRULL
11
Qn
ui
for all
Ωδ . If p ∈ P , then min{vP
p (( i=1 fi )(w)) | w ∈ R} =
Ppn ∈ P , hence x ∈ Ωδ \{0}. Let uP∈
n
n
min{ i=1 vp (fi (w))uQi | w ∈ T } = min{ i=1 vp (fi (w))ui | w ∈ S} = i=1 vp (fi (δ(p)))ui , and if p ∈
n
Q\P , then min{vp (( i=1 fiui )(w)) | w ∈ R} = 0. Let y, z ∈ Ωδ be such that x = y + z. If p ∈ P , then
Pn
Pn
Pn
Qn
xi
(fi (δ(p)))xi = i=1 vp (fi (δ(p)))yi + i=1 vp (fQ
min{vp ((Qi=1 fi )(w)) | w ∈ R} = i=1 vpQ
i (δ(p)))zi =
n
n
n
yi
xi
zi
)(w))
|
w
∈
R}.
This
implies
that
d
(
)(w))
|
w
∈
R}
+
min{v
((
f
min{v
((
f
Q
p
i=1 fi ) =
i=1 i
i=1 i Q
Qnp
n
yi
zi
dQ ( i=1 fi )dQ ( i=1 fi ), hence y = 0 or z = 0. Therefore, x ∈ A(Ωδ ).
Now we present the main result of this section.
Theorem 5.2. Let R be a factorial domain. Then Int(R) is monadically Krull.
Proof. Let K be a field of quotients of R, X an indeterminate over K, Q a system of representatives of
prime elements of R and d = dQ . Set S = R[X] and T = Int(R). It is well known that T is atomic
and completely integrally closed (see [7, Propositions VI.2.1 and VI.2.9]). By Theorem 3.6 we need to
prove that {yT | y ∈ A([[g]]T )} is finite
all g ∈ T • . Let g ∈ T • . Some a, b ∈ R• , n ∈ N, v ∈ Nn0 and
Qfor
n
a
n
f ∈ (A(S)\R) exist such that g = b i=1 fivi and fj 6'S fk for all different j, k ∈ [1, n]. By Lemma 5.1
Qn
α
fi i
αi T
i=1 fi )
it is sufficient to show that {yT | y ∈ A([[g]]T )} ⊆ {yT | y ∈ A(R), y |R d(g)} ∪ { d(Qi=1
n
| α ∈ Nn0 , α
is f -irreducible}. Let y ∈ A([[g]]T ). Then y ∈ A(T ) and y |T g l for some l ∈ N.
Case 1: y ∈ R. We have y ∈ A(R) and y |R d(g l ) = d(g)l . Therefore, y |R d(g).
Case 2: y 6∈ R. Some primitive t ∈ S and some c, e ∈ R• exist such that GCDS (c, et) = S × and y = et
c .
ed(t)
t
t
×
This implies that c |R d(t). Observe that y = ed(t)
·
,
∈
T
and
∈
T
\T
.
Consequently,
c
d(t)
c
d(t)
Qn
t
l
. There are some w ∈ S and u ∈ R• such that y w
=
g
.
Therefore,
etwbl = cual i=1 filvi
y 'T d(t)
u
Qn
lvi
and since t is primitive it follows that tQ|S
is some α ∈ Nn0 \{0} such that
i=1 fi . Hence, there
Qn
αi
α
n
Qn
fi
fi i
n
Qi=1
t 'S i=1 fiαi . This implies that y 'T d(Qi=1
αi , and thus yT =
αi T . Let β, γ ∈ N0 be such
n
n
d(
f
)
f
i=1 i
i=1 i )
Q
Q
γ
β
n
n
Qn
Qn
Qn
f i
fi i
Qi=1 iγ
∈ T and
that α = β + γ and d( i=1 fiαi ) = d( i=1 fiβi )d( i=1 fiγi ). Note that Qi=1
βi ,
n
d( n f i )
d(
Qn
β
fi i
Qi=1
β
n
d( i=1 fi i )
·
Qn
γ
f i
Qi=1 iγi
d( n
f
i=1 i )
=
Qn
α
f i
Qi=1 iαi
d( n
f
i=1 i )
∈ A(T ). Therefore,
Qn
β
fi i
Qi=1
β
n
d( i=1 fi i )
∈ T
×
i=1
or
β = 0 or γ = 0. Consequently, α is f -irreducible.
fi )
i=1
Qn
γ
fi i
Qi=1
γ
i
d( n
i=1 fi )
i
∈ T × , hence
Theorem 5.2 is also interesting from a purely factorization theoretical point of view, since it provides a
class of Krull monoids whose arithmetic is not fully understood by now. The arithmetic of the Krull
monoids involved may also differ from the arithmetic of monadic submonoids of principal orders in
algebraic number fields.
Corollary 5.3. Let R be a factorial domain. Then Int(R) is an FF-domain.
Proof. This follows from Theorems 3.6 and 5.2.
In [13] it has been shown that Int(Z) is an FF-domain. Corollary 5.3 is a generalization of this result.
By Theorem 5.2, [7, Theorem VI.1.7] and [7, Remark VI.2.10] we obtain that Int(Z) and Int(Z(p) ) for
p ∈ P are monadically Krull domains and Prüfer domains (and thus t-Prüfer domains) that are no Krull
domains.
6. Further remarks
In section four we showed that a radical factorial FF-monoid is not necessarily a Krull monoid. So far
we do not know whether every radical factorial, 1-dimensional FF-domain is a Krull domain. In this last
section we investigate special types of examples that have been introduced in [19] to construct atomic
Prüfer domains that are no Dedekind domains. We study these examples in greater detail and generality
to obtain an interesting class of examples that are not “too far away” from being examples of radical
factorial 1-dimensional FF-domains that are not Krull.
12
ANDREAS REINHART
T
Let H be a monoid. We say that H is a weakly Krull monoid if P ∈X(H) HP = H and {P ∈ X(H) | a ∈ P }
is finite for all a ∈ H • . Note that H is a Krull monoid if and only if H is a weakly Krull monoid and HP
is a Krull monoid for all P ∈ X(H). It follows from Example 4.2 that being a weakly Krull monoid is not
a monadic property (since the monoid in this example is monadically Krull, hence monadically weakly
Krull, but by [29, Proposition 2.6] it fails to be weakly Krull). By [2, Theorem 1] and [1, Theorem 5.1]
we have H is an FF-monoid if and only if H is atomic and {uH | u ∈ A(H), u |H x} is finite for all
x ∈ H • (such monoids are called IDF-monoids, for example see [23]) if and only if H is a BF-monoid
and {uH | u ∈ A(H), u |H x} is finite for all x ∈ H • . Clearly, if H is a BF-monoid, then H satisfies
the ACCP. First we start with a simple Lemma that might be of independent interest. It gives a hint
how to construct monoids H where {uH | u ∈ A(H), u |H x} is finite for all x ∈ H • , but that fail to be
FF-monoids.
Lemma 6.1. Let K be a monoid, S a submonoid of K that is an FF-monoid, T ⊆ K a submonoid of
K that is a valuation monoid and H = S ∩ T . Then {uH | u ∈ A(H), u |H x} is finite for all x ∈ H • .
Proof. Let x ∈ H • and D(x) = {u ∈ A(H) | u |H x}. Then {uS | u ∈ D(x)} ⊆ {uS | u ∈ S and
u |S x}. Since S is an FF-monoid, it follows that {uS | u ∈ D(x)} is finite. Let v, w ∈ A(H) be such
that vS = wS. We have vT ⊆ wT or wT ⊆ vT . Therefore, vH = vS ∩ vT ⊆ wS ∩ wT = wH or wH =
wS ∩ wT ⊆ vS ∩ vT = vH, hence vH = wH. Consequently, f : {uH | u ∈ D(x)} → {uS | u ∈ D(x)}
defined by f (I) = IS for all I ∈ {uH | u ∈ D(x)} is an injective map. This implies that {uH | u ∈ D(x)}
is finite.
Proposition 6.2. Let H be a monoid, K a quotient monoid of H, U a set of overmonoids
Tof H that are
FF-monoids and V a set of overmonoids of H that are valuation monoids such that H = S∈U ∪V S. Let
T
(NT )T ∈V ∈ P(U)V be such that T ∩ S∈NT S is atomic for all T ∈ V and {T ∈ V | S ∈ NT } is finite for
all S ∈ U. If {S ∈ U ∪ V | a 6∈ S × } is finite for all a ∈ H • , then H is an FF-monoid.
T
Proof. Let {S ∈ U ∪ V | a 6∈ S × } be finite for all a ∈ H • and M = U ∪ {T ∩ S∈NT S | T ∈ V}.
Claim 1: T
For all U ∈ M it followsTthat U is an FF-monoid. Let U ∈ M andTT ∈ V be such that
U = T ∩ S∈NT S. We show that S∈NT S is an FF-monoid. If NT = ∅, then S∈NT S = K, hence
T
×
∈ H • , we have {S ∈ NT | a 6∈ S × }
S∈NT S is an FF-monoid. Since {S ∈ NT | a 6∈ S } is finite for all a
T
•
×
is finite for all a ∈ K , hence {S ∈ NT | a 6∈ S } is finite for all a ∈ ( S∈NT S)• . Therefore, [2, Theorem
T
2] implies that S∈NT S is an FF-monoid. It follows by Lemma 6.1 that U is an FF-monoid.
Claim
2: For every a ∈ H • , {S ∈SM | a 6∈ S × } is finite. Let a ∈ H • . We have {T ∈ VT| a 6∈ (T ∩
T
×
×
and thus {T ∈ V | a 6∈ (T ∩ S∈NT S)× }
S∈NT S) } ⊆ {T ∈ V | a 6∈ T }∪ S∈U ,a6∈S × {T ∈ V | S ∈ NT }, T
T
is finite. Therefore, {S ∈ M | a 6∈ S × } = {S ∈ U | a 6∈ S × }∪{T ∩ S∈NT S | T ∈ V, a 6∈ (T ∩ S∈NT S)× }
is finite. T
Since H = S∈M S, it follows by claim 1, claim 2 and [2, Theorem 2] that H is an FF-monoid.
Proposition 6.3. Let K be a monoid, H a submonoid of K and ΛTa set of intermediate monoids of H
and K such that {S ∈ Λ | a 6∈ S × } is finite for all a ∈ H • and H ∩ S∈Λ S × = H × .
1. If S satisfies the ACCP for all S ∈ Λ, then H satisfies the ACCP.
2. If S is a BF-monoid for all S ∈ Λ, then H is a BF-monoid.
Proof. 1. Let S satisfy the ACCP for all S ∈ Λ. Let (ai )i∈N ∈ H N be such that ai H ⊆ ai+1 H for all
i ∈ N. Without restriction let a1 6= 0. Let A = {S ∈ Λ | a1 6∈ S × }. Since A is finite there is some r ∈ N
such that ak S = ar S for all S ∈ A and k ∈ N≥r . It is sufficient to show that ak H = ar H for all k ∈ N≥r .
Let k ∈ N≥r and T ∈ Λ. If a1 ∈ T × , then ar , ak ∈ T × , and thus aakr ∈ T × . If a1 6∈ T × , then ar T = ak T ,
T
hence aakr ∈ T × . Consequently, aakr ∈ H ∩ S∈Λ S × = H × , and thus ar H = ak H.
×
2. Let S be a BF-monoid
T for all S ∈ Λ and set M = {(S\S ) ∩ H
T | S ∈ Λ}. It follows by [15,
Proposition 1.3.2] that n∈N (S\S × )n = {0} for all S ∈ Λ. Therefore, n∈N M n = {0} for all M ∈ M.
Let a ∈ H • \H × . Then {M ∈ M | a ∈ M } ⊆ {(S\S × ) ∩ H | S ∈ Λ, a 6∈ S × }, hence {M ∈ M | a ∈ M } is
ON MONOIDS AND DOMAINS WHOSE MONADIC SUBMONOIDS ARE KRULL
13
finite. Since a 6∈ H × , there is some T ∈ Λ such that a 6∈ T × , hence (T \T × ) ∩ H ∈ {M ∈ M | a ∈ M }.
Consequently, [15, Theorem 1.3.4] implies that H is a BF-monoid.
S
Corollary 6.4. Let H be a monoid and M ⊆ s-spec(H) such that M ∈M M = H\H × and {M ∈ M |
a ∈ M } is finite for all a ∈ H • .
1. If HM satisfies the ACCP for all M ∈ M, then H satisfies the ACCP.
2. If HM is a BF-monoid for all M ∈ M, then H is a BF-monoid.
T
T
T
Proof.
| M ∈ M}. We have H ∩ S∈Λ S × = H ∩ M ∈M (HM \MM ) = M ∈M (H\M ) =
S Let Λ = {HM
H\ M ∈M M = H × . Let a ∈ H • \H × . Then {S ∈ Λ | a 6∈ S × } = {HM | M ∈ M, a ∈ M } is finite.
Consequently, the assertions follow from Proposition 6.3.
If S is an integral domain and R ⊆ S is a subring, then let clS (R) denote the integral closure of R in S.
We say that M ∈ max(S) is critical if for each finite E ⊆ M , there exists Q ∈ max(S) such that E ⊆ Q2 .
Proposition 6.5. Let R be a Dedekind domain that is not a field, K a field of quotients of R, L/K a
field extension, S = clL (R), (Li )i∈N a sequence
of intermediate fields of K and L such that Li ⊆ Li+1
S
and [Li : K] < ∞ for all i ∈ N and L = j∈N Lj . Let (Ai )i∈N ∈ P(max(R))N be such that Ai ⊆ Ai+1 ,
S
for all i ∈ N. Set A = i∈N Ai and N = {M ∈ max(S) | M ∩ R ∈ A}. For i ∈ N set Si = clLi (R) and
Ni = {M ∈ max(S) | M ∩ R ∈ Ai }.
S
1. Si is a Dedekind domain for all i ∈ N, S is a 1-dimensional Prüfer domain and S = i∈N Si .
2. Let for all i ∈ N and P ∈ max(Si ) such that P ∩ R ∈ Ai be P * Q2 for all Q ∈ max(Si+1 ). Then
for all M ∈ N we have M is not critical and if A = max(R), then
p S is an SP-domain.
Si+1
P Si+1 ∈ max(Si+1 ). Then for all
3. Let for all i ∈ N and P ∈ max(Si ) such that P ∩ R ∈ Ai be
•
a∈S
it
follows
that
{M
∈
N
|
a
∈
M
}
is
finite
and
if
A
=
max(R),
then S is weakly Krull.
S
4. Let P ∈A P = R\R× and let for all i ∈ N and P ∈ max(Si ) such that P ∩ R ∈ Ai be P Si+1 ∈
max(Si+1 ). Then S is a BF-domain and if | max(S)\N | ≤ 1, then S is an FF-domain.
5. If there is some sequence (Mi )i∈N such that Mi ∈ max(Si ) and Mi+1 ∩ Si = Mi for all i ∈ N and
{j ∈ N | Mj Sj+1 6∈ max(Sj+1 )} is infinite, then S is not a Dedekind domain.
S
Proof. 1. Clearly, S is 1-dimensional and S = i∈N Si . By the Theorem of Krull-Akizuki we have Si is a
Dedekind domain for all i ∈ N. Since L/K is an algebraic field extension we have S is a Prüfer domain.
2. Claim 1: For all j ∈ N and M ∈ Nj it follows that M ∩ Sj * (M ∩ Sk )2 for all k ∈ N≥j . Let
j ∈ N and M ∈ Nj . We show by induction on k that M ∩ Sj * (M ∩ Sk )2 for all k ∈ N≥j . Obviously,
M ∩ Sj * (M ∩ Sj )2 . Now let k ∈ N≥j be such that M ∩ Sj * (M ∩ Sk )2 . Since (M ∩ Sj )Sk ⊆ M ∩ Sk ,
there is some ideal I of Sk such that (M ∩ Sj )Sk = (M ∩ Sk )I. Since M ∩ Sj * (M ∩ Sk )2 , it follows
that I * M ∩ Sk , hence ISk+1 * M ∩ Sk+1 . We have M ∩ Sk ∈ max(Sk ), M ∩ Sk+1 ∈ max(Sk+1 ) and
(M ∩Sk )∩R = M ∩R ∈ Aj ⊆ Ak , and thus (M ∩Sk )Sk+1 * (M ∩Sk+1 )2 . Since (M ∩Sk+1 )2 is M ∩Sk+1 primary it follows that (M ∩Sj )Sk+1 = (M ∩Sk )Sk+1
∩Sk+1 )2 , hence M ∩Sj * (M ∩Sk+1 )2 .
S ISk+1 * (M
2
2
Claim 2: For every M ∈ max(S) we have M = i∈N (M ∩ Si ) . Let M ∈ max(S). “⊆”: Let x ∈ M 2 .
Pr
There exist some r ∈ N and (xi )ri=1 , (yi )ri=1 ∈ M [1,r] such that x = i=1 xi yi . There is some l ∈ N such
that xi , yi ∈ Sl for all i ∈ [1, r]. Consequently, x ∈ (M ∩ Sl )2 . “⊇”: Trivial.
Now let Q ∈ N . There is some j ∈ N such that Q ∈ Nj . Assume that there is some M ∈ max(S) such
that Q ∩ Sj ⊆ M 2 . Then Q ∩ Sj = M ∩ Sj and M ∩ R = Q ∩ R ∈ Aj , hence M ∈ Nj . It follows by
claim 2 that there exists some k ∈ N≥j such that M ∩ Sj ⊆ (M ∩ Sk )2 which is a contradiction to claim
1. Consequently, (Q ∩ Sj )S * M 2 for all M ∈ max(S). Since (Q ∩ Sj )S is a finitely generated ideal of S
we have Q is not critical.
Now let A = max(R). Then N = max(S), hence every M ∈ max(S) is not critical. It follows by 1 that
S is a 1-dimensional Prüfer domain. Consequently, S is an SP -domain by [26, Corollary 2.2].
3. Claim: For all i ∈ N we have fi : Ni → {M ∩ Si | M ∈ Ni } defined by fi (M ) = M ∩ Si is a bijective
map. Let i ∈ N. Obviously, fi is a surjective map. Let M, Q ∈ Ni be such that M ∩ Si = Q ∩ Si . We
14
ANDREAS REINHART
show by induction on j that for all j ∈ N≥i , M ∩ Sj = Q ∩ Sj . Let j ∈ N≥i . p
The assertion holds for j = i.
Sj+1
Now let M ∩ Sj = Q ∩ Sj . We have (M ∩ Sj )Sj+1 ⊆ M ∩ Sj+1 , hence
(M ∩ Sj )Sj+1 = M ∩ Sj+1 .
p
Sj+1
(Q
∩
S
)S
=
Q
∩
S
,
hence
M
∩
S
=
Q
∩
S
. Finally, it follows that
Analogously
j
j+1
j+1
j+1
j+1
S
S
M = j∈N≥i (M ∩ Sj ) = j∈N≥i (Q ∩ Sj ) = Q.
Let a ∈ S • . There is some l ∈ N such that a ∈ Sl . Obviously, there is some surjective map from
{M ∩ Sl | M ∈ N , a ∈ M } to {M ∩ R | M ∈ N , a ∈ M }. Since {M ∩ Sl | M ∈ N , a ∈ M } ⊆ {Q ∈
max(Sl ) | a ∈ Q} and {Q ∈ max(Sl ) | a ∈ Q} is finite we have {M ∩ R | M ∈ N , a ∈ M } is finite.
Therefore, there is some k ∈ N≥l such that {M ∩ R | M ∈ N , a ∈ M } ⊆ Ak . Since fk ({M ∈ Nk |
a ∈ M }) = {M ∩ Sk | M ∈ Nk , a ∈ M } ⊆ {Q ∈ max(Sk ) | a ∈ Q}, it follows by the claim that
{M ∈ N | a ∈ M } = {M ∈ Nk | a ∈ M } is finite.
Now let A = max(R). Then N = max(S) = X(S) by 1, and thus S is a weakly Krull domain.
4. It follows by 3 that {M ∈ N | a ∈ S} is finite for all a ∈ S • . By [18, Proposition 4] we have
S
×
2
2
M ∈N M = S\S . Let M ∈ N . By 2 it follows that M is not critical, hence M 6= M . Since M is
2
M -primary we have MM 6= MM . Therefore, 1 implies that SM is a valuation domain, and thus MM is
a principal ideal of SM . This implies that SM is a Dedekind domain, hence SM is an FF-domain and a
BF-domain. Consequently, Corollary 6.4.2 implies that S is a BF-domain. Now let | max(S)\N | ≤ 1. Set
U = {SM | M ∈ N } and V = {SM | M ∈ max(S)\N }. Every T ∈ U is an FF-domain and
T by 1 we have
that every T ∈ V is a valuation domain. Obviously,T
U ∪ V = {SM | M ∈ max(S)}, hence T ∈U ∪V T = S.
For T ∈ V set NT = U. Since |V| ≤ 1, we have T ∩ U ∈NT U = S is atomic for all T ∈ V. It follows that
{T ∈ U ∪ V | a 6∈ T × } is finite for all a ∈ S • , and thus Proposition 6.2 implies that S is an FF-domain.
5. Let (Mi )i∈N be such that Mi ∈Smax(Si ) and Mi+1 ∩ Si = Mi for all i ∈ N and {j ∈ N | Mj Sj+1 6∈
max(Sj+1 )} is infinite. Let M = i∈N Mi . Observe that M ∈ max(S). Assume that S is a Dedekind
domain, then there is some finite E ⊆ M such that M = (E)S . There is some i ∈ N such that E ⊆ Mi .
There is some j ∈ N≥i such that Mj Sj+1 6∈ max(Sj+1 ), and thus there are some Q, Q0 ∈ max(Sj+1 ) such
that Mj Sj+1 ⊆ QQ0 . This implies that M = QS = Q0 S and M 2 = QSQ0 S = M , hence M = {0}, a
contradiction.
Proposition 6.6. Let R be a Dedekind domain such that max(R) is infinite, K a field of quotients of R,
N
(Ai )i∈N , (Bi )i∈N , (Ci )i∈N ∈ P(max(R))
such
Ci are finite and Ai ⊆ Ai+1 , Bi ⊆ Bi+1 , Ci ⊆
S
S that Ai , Bi , S
Ci+1 for all i ∈ N. Set A = i∈N Ai , B = i∈N Bi and C = i∈N Ci . Assume that A∩B = A∩C = B∩C = ∅
and let R/M be finite for all M ∈ max(R). Then there exists some sequence (Li )i∈N of extension fields
of K such that L1 = K, Li ⊆ Li+1 , [Li : K] < ∞ and Si = clLi (R) for all i ∈ N and such that the
following conditions are satisfied:
1. For all i ∈ N and M ∈ max(Si ) such that M ∩ R ∈ Ai we have M Si+1 ∈ max(Si+1 ).
2. For all i ∈ N and M ∈ max(Si ) such that M ∩ R ∈ Bi we have M Si+1 6∈ max(Si+1 ) and M * Q2
for all Q ∈ max(Si+1 ).
3. Forpall i ∈ N and M ∈ max(Si ) such that M ∩ R ∈ Ci we have M Si+1 6∈ max(Si+1 ) and
Si+1
M Si+1 ∈ max(Si+1 ).
Proof. This follows by induction from [16, Theorem 42.5].
By [29, Example 4.3] there is some Dedekind domain R such that max(R) is countable, R/M is finite for
all M ∈ max(R) and Pic(R) isStorsion-free. Let M : N0 → max(R) be a bijection. Note that since Pic(R)
is torsion-free we obtain that M ∈max(R)\{M0 } M = R\R× (since every non-unit of R is contained in at
least two different maximal ideals of R). For j ∈ N set Aj = {Mi | i ∈ [1, j]}.
S
First set Bj = {M0 } and Cj = ∅ for all j ∈ N. Let (Li )i∈N be the sequence in Proposition 6.6, L = i∈N Li
and S = clL (R). Then S is an SP-domain that is a BF-domain but not Krull by Proposition 6.5.S
Next set Bj = ∅ and Cj = {M0 } for all j ∈ N. Let (Li )i∈N be the sequence in Proposition 6.6, L = i∈N Li
and S = clL (R). Then S is a completely integrally closed FF-domain that is a weakly Krull domain but
not a Krull domain by Proposition 6.5.
ON MONOIDS AND DOMAINS WHOSE MONADIC SUBMONOIDS ARE KRULL
15
Proposition 6.7. Let R be a Prüfer domain, K a field of quotients of R, L/K an algebraic field extension
and S = clL (R).
1. If for all intermediate fields K ⊆ M ⊆ L such that [M : K] < ∞ it follows that Pic(clM (R)) is a
torsion group, then Pic(S) is a torsion group.
2. If for all a ∈ L and n ∈ N there is some b ∈ L such that bn = a, then Pic(S) is torsion-free.
m
[1,m]
Proof. 1. Let I P
be an invertible ideal of S. Then there are some m ∈ N and some sequence
Pm (ai )i=1 ∈ I
m
such that I = i=1 ai S. Set M = K({ai | i ∈ [1, m]}), T = clM (R) and J = i=1 ai T . Note that
{ai | i ∈ [1, m]} ⊆ M ∩ S = T , and thus J is a non-zero finitely generated ideal of T . Since T is a Prüfer
domain we have J is an invertible ideal of T . Since Pic(clM (R)) is a torsion group, there are some n ∈ N
and a ∈ T such that J n = aT . Therefore, I n = J n S = aS.
2. Let I be an invertible ideal of S, n ∈ N and a ∈ S such that I n = aS. There is some b ∈ L such
that bn = a. Observe that b ∈ S and I n = bn S. Let M ∈ max(S). Since S is a Prüfer domain it
follows that SM is a valuation domain, hence there is some c ∈ S such that IM = cSM . This implies that
×
n
cn SM = IM
= bn SM , and thus there exists some ε ∈ SM
such that cn = εbn . Since SM is a valuation
domain it follows that bSM ⊆ cSM or cSM ⊆ bSM . Case 1: bSM ⊆ cSM . There exists some ν ∈ SM
×
such that b = cν. This implies that bn = cn ν n = εbn ν n , hence 1 = εν n . Consequently, ν ∈ SM
, and thus
IM = cSM = cνSM = bSM . Case 2: cSM ⊆ bSM . There is some ν ∈ SM such that c = bν. We have
×
cn = bn ν n = ε−1 cn ν n , hence ν n = ε. This implies that ν ∈ SM
, and thus IM = cSM = bνSM = bSM .
Therefore, IQ = bSQ for all Q ∈ max(S), hence I = bS.
Let H be a monoid. So far we said little about the additional property that popped up in Theorem 3.6.3
(i.e. that for every a ∈ H • , A([[a]]) ⊆ Dk (a) for some k ∈ N). Note that this additional property is
equivalent to the notion of being pseudo-IDPF introduced in [23]. Since we are interested in studying
monadically Krull monoids and their specializations, we investigate how to control the r-ideal class group
of an r-SP-monoid to obtain this additional property. This is reasonable since there are non-trivial
situations using the construction in Proposition 6.5 where SP-domains that are BF-domains can show
up (as pointed out before). Moreover, Proposition 6.7 indicates that the class group of domains in this
construction can behave nicely. If G is an abelian group, then let exp(G) be the exponent of G (i.e. if 1
is the identity of G, then exp(G) = inf({n ∈ N | xn = 1 for all x ∈ G})). The group G is called bounded
if exp(G) < ∞.
Proposition 6.8. Let H be a monoid and r a finitary ideal system on H such that H is an r-SP-monoid.
1. If Cr (H) is finite, then for all a ∈ H • , A([[a]]) ⊆ Dk (a) for some k ∈ N.
2. If H is an FF-monoid and Cr (H) is bounded, then for all a ∈ H • , A([[a]]) ⊆ Dk (a) for some k ∈ N.
3. If H is r-Prüfer and Cr (H) is bounded, then for all a ∈ H • , A([[a]]) ⊆ Dk (a) for some k ∈ N.
Proof. 1. Let a ∈ H • . Set k = |Cr (H)|. We prove that A([[a]]) ⊆ Dk (a). Let u ∈ A([[a]]). Q
There are
s
some l, s ∈ N and some sequence (Ii )si=1 of proper radical r-ideals of H such that al ∈ uH = ( i=1 Ii )r .
l
s
Observe that a ∈ Ii for all i ∈ [1, s], hence a ∈ Ii for all i ∈ [1, s]. This implies that a ∈ uH. If s ≤ k,
k
then
6 E ⊆ [1, s] such
Q a ∈ uH, and thus u ∈ Dk (a). Now
Qs let s > k.QThere is some ∅ =
Q that |E| ≤ k and
( i∈E Ii )r is principal. Since uH = ( i=1 Ii )r ⊆ ( i∈E Ii )r $ H, we have uH = ( i∈E Ii )r . Therefore,
Q
ak ∈ a|E| H ⊆ ( i∈E Ii )r = uH. Consequently, u ∈ Dk (a).
2. Let H be an FF-monoid, Cr (H) bounded and a ∈ H • . Set l = exp(Cr (H)), M = {I | I is an r-invertible
radical r-ideal of H, a ∈ I} and N = {bH | b ∈ H, al ∈ bH}. Let f : M → N be defined by f (I) = (I l )r
for all I ∈ M. If I ∈ M, then there is some b ∈ H such that (I l )r = bH. Set J = aI −1 . Then J ∈ Ir (H)
l
and aH = (IJ)r . This implies that al ∈ al H = (I l J l )r = b(Jp
)r ⊆ bH,p
and thus p
f is well-defined.
Now
p
l
l
let I, J ∈ M be such that f (I) = f (J). It follows that I = (I )r = f (I) = f (J) = (J )r = J.
Therefore, f is injective. Since H is an FF-monoid we have |M| ≤ |N | < ∞. Set k = l|M|. We show
that A([[a]]) ⊆ Dk (a). Let u ∈ A([[a]]). There are some m, n ∈ N, some sequence (αiQ
)ni=1 ∈ N[1,n] and some
n
n
m
sequence (Ii )i=1 of distinct proper radical r-ideals of H such that a ∈ uH = ( i=1 Iiαi )r . Note that
16
ANDREAS REINHART
α
Ii ∈ M for all i ∈ [1, n], hence n ≤ |M|. If αj > l for some j ∈ [1, n], then uH ⊆ (Ij j )r ⊆ (Ijl )r ⊆ I $ H,
α
α
and thus uH = (Ijl )r = (Ij j )r which implies that Ij = H, a contradiction. Therefore, al ∈ (Ij j )r for all
Q
n
j ∈ [1, n]. It follows that ak ∈ ( i=1 Iiαi )r = uH, hence u ∈ Dk (a).
3. Let H be an r-Prüfer monoid, Cr (H) bounded and a ∈ H • . Set m = exp(Cr (H)) and k = 2m. It is
sufficient to show that A([[a]]) ⊆ Dk (a). Let u ∈ A([[a]]). By [29, Proposition 3.9], [29, Theorem 3.13] and
[29, Theorem 3.3.2] there are some l ∈ N and some ascending sequence (Ii )li=1 of proper radical r-ideals
Ql
of H such that uH = ( i=1 Ii )r . Set F = Il . Then F is r-invertible. Clearly, uH ⊆ (F l )r and there is
some b ∈ H\H × such that (F m )r = bH. Assume that l > k. We have uH ⊆ (F l )r ⊆ (F k )r = b2 H ⊆ bH.
Since u ∈ A(H), this implies that uH = b2 H = bH, hence b ∈ H × , a contradiction. Therefore, l ≤ k,
Ql
and thus ak ∈ al H ⊆ ( i=1 Ii )r = uH. Consequently, u ∈ Dk (a).
Acknowledgement. We want to thank A. Geroldinger, F. Halter-Koch, F. Kainrath and the referee for
their comments and suggestions.
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Institut für Mathematik und wissenschaftliches Rechnen, Karl-Franzens-Universität, Heinrichstrasse 36, 8010
Graz, Austria
E-mail address: [email protected]