STRUCTURE ON THE SET OF CLOSURE OPERATIONS OF A
COMMUTATIVE RING
JANET C. VASSILEV
Abstract. We investigate the algebraic structure on the set of closure operations of a
ring. We show the set of closure operations is not a monoid under composition for a
discrete valuation ring. Even the set of semiprime operations over a DVR is not a monoid;
however, it is the union of two monoids, one being the left but not right act of the other.
We also determine all semiprime operations over the ring K[[t2 , t3 ]].
1. Introduction
Let I → Ic be an operation on the set of ideals of a ring R. Consider the following
properties where I and J are ideals and b is a regular element:
(a) I ⊆ Ic
(b) If I ⊆ J, then Ic ⊆ Jc .
(c) (Ic )c = Ic .
(d) Ic Jc ⊆ (IJ)c
(e) (bI)c = bIc
If I → Ic satisfies (a)-(c) above, we call I → Ic a closure operation. If I → Ic is a closure
operation and also satisfies (d) above, we call I → Ic a semiprime operation. If I → Ic is
semiprime and also satisfies (e), then we say I → Ic is a prime operation.
There are many well known closure operations defined on a commutative ring, such as:
integral closure, tight closure if the ring contains a field [HH], ∆-closure [Ra], basically
full closure [HRR], etc. It is known that all of these closure operations are contained in
the integral closure, excluding the ∆-closure. However, if ∆ doesn’t contain any ideals
which are contained in a minimal prime, then the ∆-closure is also contained in the integral
closure. Otherwise, the relationships between these other closures is not as well understood. Knowing the structure on the set of closure operations may shed some light into this
relationship.
Abstractly, closure operations can be thought of subsets of the monoid of maps from the
set of ideals of a ring to itself, MI = {f : I → I|I is the set of ideals of R} satisfying the
above properties. For example, CR is the set of maps satisfying (a)-(c), SR is the set of
maps satisfying (a)-(d) and PR is the set of maps satisfying (a)-(e). CR , SR and PR are all
partially ordered sets, but otherwise these sets are in general poorly behaved. In Section
2, we will give examples showing that CR is not even a monoid in the nice case that R is a
discrete valuation ring. Then in Section 3 we show that SR for a discrete valuation ring R is
almost a monoid, i.e. SR is the union of two submonoids of MI one a left but not a right act
Key words and phrases. closure operation, semiprime operation, integral closure, tight closure, monoid.
1
2
JANET C. VASSILEV
of the other. Also we show that PR is a monoid. We are also able to extend our results on
semiprime and prime operations over a Dedekind domain. In Section 4, we consider closure
operations over the semigroup ring K[[t2 , t3 ]] and determine all the semiprime operations
over K[[t2 , t3 ]].
I would like to thank Dave Rush, for giving some inspirational talks on closure operations
in the commutative algebra seminar at UC Riverside and for his many suggestions for
strengthening this work. I would also like to thank Bahman Engheta and Jooyoun Hong
for being a sounding board for my ideas and Hwa Young Lee for pointing out an error in
the work over K[[t2 , t3 ]].
2. Preliminaries
Recall that (S, ◦) is a semigroup if ◦ is an associative binary operation on S. We say
that a semigroup (S, ◦) is a monoid if there is a unique identity element e in S such that
es = se = e for all s ∈ S.
Let R be a commutative ring. Let I = {I ⊆ R|I an ideal of R}. Let MI = {f : I → I}.
MI is clearly a monoid under composition of maps with identity, the identity map e : I → I
and function composition is associative. CR will be the subset of MI consisting of closure
operations. Hence the fc in CR are the set of maps satisfying the following three properties:
(a) fc (I) ⊇ I, (b) fc preserves inclusions in R, and (c) fc ◦ fc = fc . SR will be the set
of semiprime operations of R, i.e. SR are the maps in CR which also satisfy fc (I)fc (J) ⊆
fc (IJ). PR will be the set of prime operations of R, i.e. maps in SR which also satisfy the
(e) fc (bI) = bfc (I). We note that if CR , SR or PR are monoids, by property (c), they will
be band monoids.
Definition 2.1. A monoid is a band monoid if every element is idempotent.
We will say fc1 ≤ fc2 for two different closure operations if fc (I) ⊆ fd (I) for all I ∈ I.
Proposition 2.2. CR ,SR and PR are partially ordered sets.
The proof is straightforward as the ideals of R are are partially ordered under containment.
Now let us consider the algebraic structure of CR , R a commutative ring. Unfortunately,
CR is not a submonoid under composition even for a discrete valuation ring.
Example 2.3. CR , where (R, P ) is a discrete valuation ring is not a monoid. The ideals
of R have the form P i for all i ≥ 0 and (0). Let fn : I → I and gn : I → I be defined as
follows
(
(
P i for i ≤ n
R for i ≤ n
i
i
fn (P ) =
and gn (P ) =
n
P for i > n.
P n for i > n.
and fn (0) = (0) = gn (0). If m > n, then
fn ◦ gm (P i ) =
(
R for i < m
P n for i ≥ m.
This fails property (c) as (fn ◦ gm ) ◦ (fn ◦ gm )(P i ) = R for all i.
STRUCTURE ON THE SET OF CLOSURE OPERATIONS OF A COMMUTATIVE RING
3
We will see in the next section that gn in the above example is not a semiprime operation,
i.e., semiprime operations are not allowed to have any finite jumps.
In Example 2.3 we see that the maps fn and gn are bounded maps on the ideals of R.
This prompts the following definition for closure operations of commutative rings:
Definition 2.4. We say a closure operation fc is bounded on a commutative ring R if for
every maximal ideal m of R, there is an m-primary ideal I such that for all m-primary
J ⊆ I, fc (J) = I. If this is not the case, we will say that fc is an unbounded closure
operation.
3. Algebraic structure on SR and PR when R is a Dedekind domain
It seems unlikely that SR and PR are submonoids of MI for a general commutative ring
R, but in the case that R is a discrete valuation ring, PR is the trivial submonoid of MI
and SR decomposes into the union of two submonoids whose only common element is the
identity. We use the following definition to explain their relationship.
Definition 3.1. Let S be a monoid and A any set, then we say A is a left (right) Sact if there is a map δ : S × A → A (δ : A × S → A) satisfying δ(st, a) = δ(s, δ(t, a))
(δ(a, st) = δ(δ(a, s), t)) for every a ∈ A and s, t ∈ S and δ(e, a) = a (δ(a, e) = a) for all
a ∈ A where e is the identity of S.
Proposition 3.2. When (R, P ) is a discrete valuation ring, SR can be decomposed into the
union of two submonoids
¯
(
(
)
¯
i for 0 ≤ i < m
[
P
¯
fm ∈ MI ¯fm (P i ) =
M0 = {e}
and fm (0) = (0)
¯
P m for i ≥ m
and
Mf = {e}
[
(
gm
¯
(
¯
P i for 0 ≤ i < m
¯
∈ MI ¯gm (P i ) =
¯
P m for i ≥ m
)
and gm (0) = P m
where Mf is a left M0 -act but not a right M0 -act under composition.
Before proving the theorem, we need the following lemma:
Lemma 3.3. Let fc be a semiprime operation on the DVR (R, P ), then fc is not constant
for P i on a finite interval m ≤ i ≤ n for m < n.
Proof: The ideals of R have the form P i and they are totally ordered. Being a closure
operation, fc (P i ) = P j ⊇ P i , where j ≤ i. Thus fc must be increasing on the ideals of R.
Suppose fc is constant for P i , where m ≤ i ≤ n, m < n. If fc (P i ) = P j for m ≤ i ≤ n
where j > m, then P m * fc (P m ) = P j . Thus fc (P i ) = P j for some j ≤ m.
Since fc is increasing, fc (P j ) = P k , k ≤ j. If k < j, then fc ◦ fc (P i ) = P k 6= P j
contradicting the fact that fc was a closure operation. Again using that fc is increasing
we see that fc (P i ) = P j for all j ≤ i < m. Thus we may assume that we have chosen a
maximal subinterval where fc is constant and j = m.
4
JANET C. VASSILEV
As fc is a semiprime operation, then fc (P i )fc (P j ) ⊆ fc (P i+j ) for all i and j; however,
fc (P )fc (P n ) = P n+1 properly contains fc (P n+1 ) = P n+1 . Thus fc cannot be constant on a
finite interval.
¤
Proof of 3.2: The ideals of a discrete valuation ring (R, P ) are either of the form P i for
i ≥ 0 or (0) and they are totally ordered R ⊇ P ⊇ P 2 ⊇ · · · P m ⊇ · · · ⊇ (0).
By Lemma 3.3, we know that any semiprime operation on R is not constant on a finite
interval m ≤ i ≤ n, m < n.
A semiprime closure operation, fc , may be constant for P i on an infinite interval i ≥ m.
Note that
(
P k for k < m
fc (P k ) =
P m for k ≥ m.
Now
 j+k
P
for j, k < m



P j+m for j < m, k ≥ m
fc (P j )fc (P k ) =

P m+k for j ≥ m, k < m


 2m
P
for j, k ≥ m
and
(
fc (P
j+k
)=
P j+k for j + k < m
P m for j + k ≥ m.
Hence fc (P j )fc (P k ) ⊆ fc (P j+k ).
Thus, a semiprime operation on R is either the identity map e or of the forms fm (P i ) =
gm (P i ) for all i ≥ 0 and fm (0) = (0), but gm (0) = P m .
Clearly fm ◦ fn = fmin(m,n) and gm ◦ gn = gmin(m,n) both imply that the corresponding
sets of semiprime operations in SR , M0 and Mf are submonoids of MI . That Mf is a left
M0 -act can be seen by fn ◦ gm = gmin(m,n) . However, for m > n, gm ◦ fn (0) = P m and
(gm ◦ fn ) ◦ (gm ◦ fn ) = gn which
S implies gm ◦ fn is not a closure operation. Thus, Mf is not
a right M0 -act and SR = M0 Mf is not a submonoid.
¤
For every n ≥ 0, Mn = {e} ∪ {fn } ∪ {gn } also form finite submonoids of MI contained in
SR , inter-relating M0 and Mf .
Proposition 3.4. The only element of PR when (R, P ) is a discrete valuation ring is the
identity.
Proof: Let (b) = P . If fc is prime, then bfc (P i ) = fc (bP i ) = fc (P i+1 ). Note if fc was
either fm or gm in the above proof, then bfc (P m ) = bP m ( P m = fc (P m+1 ) = fc (bP m ).
This contradicts the assumption of primeness. Thus PR = {e}.
¤
If R is a Dedekind domain which is not necessarily local then for every maximal ideal
m in R, Rm is a discrete valuation ring. We know the structure SRm , and can build the
structure of SR from SRm .
Given
i ∈ Λ. Consider the monoid given
Q a Dedekind domain R with maximal ideals Pi ,Q
by
N0 , the internal direct product of N0 . For φ ∈
N0 , φ(i) = 0 for all but finitely
i∈Λ
i∈Λ
many i. Suppose φ(ij ) = mj 6= 0 for i1 , i2 , . . . is and φ(i) = 0 all other i. This φ corresponds
1
2
s
to the ideal Pim
Pim
· · · Pim
.
s
1
2
STRUCTURE ON THE SET OF CLOSURE OPERATIONS OF A COMMUTATIVE RING
5
As the non-negative integers play a major role in identifying the
Q semiprime operations
in a discrete valuation, certain subsets of the semigroup NΛ
=
N0 will determine the
0
i∈Λ
semiprime operations of a Dedekind domain with maximal ideals Pi , i ∈ Λ. All the ideals
1
r
in a Dedekind domain are finite products of the Pi , i.e. I = Pim
· · · Pim
.
r
1
To determine these subsets, first consider the semilocal principal ideal domain R with two
maximal ideals P and Q, the ideals of R are P i Qj i, j ≥ 0 which corresponds to the lattice
point (i, j) in N20 . Suppose that for some semiprime operation fc defined on (R, P, Q),
(
(
i for i < m
Qj for j < n
P
j
.
and
f
(Q
)
=
fc (P i ) =
c
P n for j ≥ n
P m for i ≥ m
As fc is semiprime, we know that
fc (P i )fc (Qj ) ⊆ fc (P i Qj ) ⊆ fc (P i ) ∩ fc (Qj )
as P i Qj ⊆ P i and P i Qj ⊆ Qj . Thus
 m n
P Q if i ≥ m and j ≥ n



P m Qj if i ≥ m and 0 ≤ j < n
fc (P i Qj ) =

P i Qn if 0 ≤ i < m and j ≥ n


 i j
P Q if 0 ≤ i < m and 0 ≤ j < n.
We define the identity rectangle B of a semiprime operation fc on the lattice (R, P, Q) to
j ) = P i Qj .
be the (i, j) such that fc (P i QQ
Recall, the elements φ ∈
N0 , φ(i) = 0 for all but finitely many i. Suppose φ(ij ) =
i∈Λ
Q
mj 6= 0 for i1 , i2 , . . . is and φ(i) = 0 all other i. Each φ of
N0 is associated to the unique
i∈Λ
1
2
s
ideal of the Dedekind domain R. Note φ as above, corresponds to the ideal Pim
Pim
· · · Pim
.
s
1
2
For simplicity, we will denote the ideal corresponding to φ by I(φ). Similarly we can define
an identity Λ-box for R with maximal ideals indexed by Λ.
Definition 3.5. The identityQΛ-box of the semiprime operation fc over a Dedekind Domain
R, BΛ , is the set of all φ ∈
N0 , fc (I(φ)) = I(φ).
i∈Λ
For simplicity we will denote φji to be the element of
Q
N0 such that φ(i) = j and
i∈Λ
···jr
φ(λ) = 0 for all λ 6= i. All elements are of the form φji11 + φji22 · · · + φjirr := φji11ij22···i
distinct ik .
r
Note that in identity Λ-box, BΛ of fc could be bounded if for every i ∈ Λ there is a finite m
j
m
with
Qfc (I(φi )) = I(φi ) for j ≥ m. In fact all semiprime operations on the principal ideals
of
N0 have the form
i∈Λ

j1 j2 ···jr
j1 ,j2 ,...,jr

I(φi1 i2 ···ir ) if φi1 ,i2 ,...,ir ∈ BΛ
···jr
,...,jr
···kr
fBΛ (I(φji11ij22···i
)) = I(φki11ik22···i
) if φji11,i,j22,...,i
∈
/ BΛ and kl = ml
r
r
r


for some l and kh = jh for all h with kh ≤ mh if mh exists.
6
JANET C. VASSILEV
If BΛ and CΛ are any two identity Λ-boxes. Clearly, BΛ ∩ CΛ is also a identity Λ-box
and fBΛ ◦ fBΛ = fBΛ ∩CΛ .
Q
As the semiprime operations of a Dedekind domain correspond to elements of
N0 ∪{∞}
i∈Λ
under partial ordering, when BΛ is bounded with a finite number i ∈ Λ with mi 6= 0 there
are two types of semiprime operations fBΛ and gBΛ . The only difference is that fBΛ (0) = (0)
2
r
1
Pim
· · · Pim
.
and gBΛ 0(0) = Pim
r
1
2
Let us define two subsets of MI :
S
• Mf = {e} S{gBΛ ∈ MI } the set of finite closure operations.
• M0 = {e} {fBΛ ∈ MI } the set of closure operations for which the zero ideal is
closed.
Suppose now that BΛ and CΛ are two identity Λ-boxes with both BΛ and CΛ bounded
and finite. Then BΛ ∩ CΛ is also bounded and finite and is also a identity Λ-box and
gBΛ ◦ gCΛ = gBΛ ∩CΛ . This shows that Mf is a submonoid of MI .
Lastly, suppose that BΛ and CΛ are two identity Λ-boxes with CΛ bounded and finite.
Then BΛ ∩ CΛ ( CΛ is also bounded as above and is also a identity Λ-box. Note that,
fBΛ ◦ gCΛ = gBΛ ∩CT
but gCΛ ◦ fBΛ 6= gBΛ ∩CΛ since
Λ
gCΛ ◦ fBΛ (0) =
Pim where φm
i ∈ CΛ 6= BΛ ∩ CΛ which is not a closure operation. This
i∈Λ
shows that Mf is a left M0 -act, but not a right M0 -act.
We have just proved:
Proposition 3.6. When RSis a Dedekind domain, SR can
S be decomposed into the union of
two submonoids M0 = {e} {fBΛ ∈ MI } and Mf = {e} {gBΛ ∈ MI } where Mf is a left
M0 -act but not a right M0 -act under composition.
Proposition 3.7. The only element of PR when R is a Dedekind domain is the identity.
Proof: Suppose (bi ) = Pi . If fc is prime, then bi fc (Pij ) = fc (bi Pi )j ) for all xi . Note if
fc was either fBΛ or gBΛ then bi fc (Pim ) = Pim+1 ⊇ Pim = fc (bi Pim ). This contradicts the
assumption of primeness. Thus PR = {e}.
¤
4. SR and PR when R = K[[t2 , t3 ]]
Although K[[t2 , t3 ]] is a local ring, the ideal structure in K[[t2 , t3 ]] is not totally ordered
as in the case of a discrete valuation ring. All ideals in K[[t2 , t3 ]] are either generated by
one element tn + atn+1 where a ∈ K or two elements of the form (tn , tn+1 ). I would like
to thank Hwa Young Lee for pointing out that I was ignoring the ideals (ti + ati+1 ), with
a 6= 0 in a previous version of this paper. She shared with me some of the ideas from her
developing thesis including some theorems which she proved which can be summed up in
the following proposition. The proof here is my own.
Proposition 4.1. The ideals of R = K[[t2 , t3 ]] can be either be expressed as a principal
ideal in the form (tn + atn+1 ), a ∈ K n ≥ 2 or as a two generated ideal (tn , tn+1 ) for n ≥ 2.
Proof: Suppose f ∈ R and (f ) 6= R. Then f is not a unit. Thus f = tn + a1 tn+1 +
a2 tn+2 + · · · for n ≥ 2. We will show that tm ∈ (f ) for m ≥ n + 2. Hence, tn + a1 tn+1 ∈ (f ).
Similarly, tm ∈ (tn + a1 tn+1 ) for m ≥ n + 2. Hence, f ∈ (tn + a1 tn+1 ).
STRUCTURE ON THE SET OF CLOSURE OPERATIONS OF A COMMUTATIVE RING
7
Let g ∈ K[[t]]. Note that tm−n g ∈ K[[t2 , t3 ]]. Hence, if g is a unit in K[[t]], then tm−n g ∈
K[[t2 , t3 ]] also. In K[[t]], f = tn (1 + a1 t + a2 t2 + · · · ) = tn g. Note that tm−n g −1 f = tm .
Similarly tm ∈ (tn + a1 tn+1 ). Thus (tn + a1 tn+1 ) = (f ). Hence, all principal ideals of
K[[t2 , t3 ]] have the form (tn + atn+1 ).
Suppose, I is not principal. As tm ∈ (tn + atn+1 ) for m ≥ n + 2, then I can be generated
by at most 2 elements of the form (tn + atn+1 , tm + btm+1 ) where m = n or m = n + 1.
If m = n, then tn+1 ∈ I which also implies that tn ∈ I. Hence I = (tn , tn+1 ).
If m = n + 1, then tn+2 ∈ (tn + atn+1 ) ⊆ I as in the principal case above. However,
n+1
t
= tn+1 + btn+2 − btn+2 ∈ I and once again tn ∈ I. Hence, I = (tn , tn+1 ).
¤
In fact the ideals are woven in the following way:
(t3 + at4 )
mH
(t3 , t4 )
HH
HH
HH
HH
H
q
qqq
q
q
q
qqq
MMM
MMM
MMM
M
(t5 + at6 )
q
qqq
q
q
q
qqq
MMM
MMM
MMM
M
(t7 + at8 ) · .
p
ppp
p
p
pp .
ppp
2
3
(t5 , t6 )
(t7 , t8 )
m4 · · ·
q m MMMM
q m MMMM
p
q
q
p
q
q
MMM
MMM
pp
qq
qq
MMM
MMM
ppp
qqq
qqq
p
q
q
M
M
p
q
q
M
M
qq
qq
pp
(t2 + at3 )
(t4 + at5 )
(t6 + at7 )
where each line in the above diagram indicates ⊇.
In the case of a discrete valuation ring (R, P ), the integral closure was the identity map
on ideals of R. For K[[t2 , t3 ]], the integral closure of ideals of the form (ti + ati+1 ), is
(ti + ati+1 ) = (ti , ti+1 ) whereas the ideals of the form (ti , ti+1 ) are all integrally closed.
Looking at the above diagram, we see that the chain of ideals in the center are all integrally
closed. However, the principal ideals are not. Clearly there are now many more closure
operations for K[[t2 , t3 ]]. In fact, the semiprime operations which are not bounded abound.
To shorten the expressions appearing in the proofs we will denote the principal ideals
Pi,a := (ti + ati+1 ) and Mi := (ti , ti+1 ).
(
Mi if I = Pi,a
2
3
int
Proposition 4.2. In K[[t , t ]], for all i ≥ 2 and all a ∈ K, the map fi (I)
I if I 6= Pi,a
is a closure operation which is not semiprime.
Proof: Clearly fiint (I) ⊇ I for all I and if I ⊆ J, fiint (I) ⊆ fiint (J). As fiint (I) = I and
fiint ◦ fiint (Pi,a ) = fiint (Mi ) = Mi = fiint (Mi )
then fiint is a closure operation.
As Mj Pk,a = Mj+k , the only ideals which are factors Pm,a are of the form Pj,b , j ≤ m − 2
and b ∈ K. If j + k = m with j, k ≥ 2, then
(
Pm,a+b if j 6= i and k 6= i
fiint (Pj,a )fiint (Pk,b ) =
Mm if j = i or k = i.
If m ≥ i + j, j ≥ 2, fiint (Pm,a+b ) = Pm,a+b and Mm * Pm,a+b . Thus fiint is not a semiprime
operation.
¤
8
JANET C. VASSILEV
We observe in the proof, that if we want such a closure operation which maps Pi,a to
Mi to be semiprime we also need Pm,a to map to Mm for m ≥ i + 2. Hence, we have the
following:
Corollary 4.3. Let S 6= ∅ and T (
, possibly empty, be subsets of the field K. Over K[[t2 , t3 ]]
/ S or I = Pi+1,b , b ∈
/T
int (I) I if I ⊇ Mi+1 , I = Pi,a , a ∈
are
for all i ≥ 2, the maps fi,S,T
I if I ⊇ Pi,a , a ∈ S or I ⊆ Pi+1,b , b ∈ T
semiprime operations.
int are also closure operations and from the proof of above, they are
Proof: Clearly fi,S,T
semiprime.
¤
Lemma 4.4. If fc is a semiprime operation on K[[t2 , t3 ]] and Mj = fc (Mj+2 ) for some j,
then fc is a bounded semiprime operation.
Proof: As
Mj ⊇ Mj+1 ⊇ Mj+2 ,
then
Mj = fc (Mj+2 ) ⊆ fc (Mj+1 ) ⊆ fc (Mj ) = Mj .
Note that if k ≥ 2,
fc (Mj+k ) ⊇ fc (P2,0 )fc (Mj+k−2 ) ⊇ P2,0 fc (Mj+k−2 ) = Mj+2
by downward induction. But
fc (Mj+2 ) = Mj implies fc (Mj+k ) = Mj
for k ≥ 2.
Note also that
Mj ⊇ Pj,a ⊇ Mj+2
implies
Mj = fc (Mj+2 ) ⊆ fc (Pj,a ) ⊆ fc (Mj ) = Mj .
Now for k ≥ 2,
fc (Pj+k,a ) ⊇ fc (P2,0 )fc (Pj+k−2,a ) ⊇ P2,0 fc (Pj+k−2,a ) = Mj+2
by downward induction. Again
fc (Mj+2 ) = Mj implies fc (Pj+k,a ) = Mj
for k ≥ 2.
Thus fc is a bounded semiprime operation.
¤
Lemma 4.5. If fc is a semiprime operation on K[[t2 , t3 ]] and fc (Mj ) = fc (Mj+2 ) for some
j, then fc is a bounded semiprime operation.
Proof: We can break the proof down into the following two cases:
(1) fc (Mj ) = Mk , k ≤ j and
(2) fc (Mj ) = Pk,a , some a ∈ K and k ≤ j − 2
STRUCTURE ON THE SET OF CLOSURE OPERATIONS OF A COMMUTATIVE RING
9
In case (1),
Mk = fc (Mk ) ⊇ fc (Mk+1 ) ⊇ fc (Mk+2 ) ⊇ fc (Mj+2 ) = Mk .
By Lemma 4.4 fc is bounded.
In case (2),
fc (Mj+2 ) = fc (Pj,b ) = fc (Mj ) = fc (Pk,a ) = Pk,a ,
for all b ∈ K. Also
fc (Mj+2 ) = fc (Mj+1 ) = fc (Mj ) = fc (Pk,a ) = Pk,a .
We will see that
fc (Mj+i ) = Pk,a
for i ≥ 3.
fc (Mj+3 ) = fc (P2,0 Mj+1 ) ⊇ fc (P2,0 )fc (Mj+1 ) ⊇ Pk+2,a .
As Pk+2,a ⊇ Pj,b , fc (Pj,b ) = fc (Pk+2,a ) = Pk,a . Thus Pk,a = fc (Mj+3 ). For the same
reason if fc (Mj+i ) = Pk,a , then fc (Mj+i+1 ) = Pk,a . We also see that for i ≥ 2 and b ∈ K,
fc (Pj+i,b ) = Pk,a as
Mj+i ⊇ Pj+i,b ⊇ Mj+i+2
Hence, fc is bounded.
¤
Lemma 4.6. If fc is a semiprime operation on K[[t2 , t3 ]] and fc (Mj ) = fc (Mj+1 ) for some
j, then fc is a bounded semiprime operation.
Proof: Note that if
I = fc (Mj ) ⊇ Mj−1 ⊇ Mj
then
I = fc (Mj−1 ) = fc (Mj+1 ).
By Lemma 4.5 we can conclude that fc is bounded. That leaves us with the cases:
(1) fc (Mj ) = Mj and
(2) fc (Mj ) = Pj−2,a for some a ∈ K.
In case (1), consider
2
fc (M2j+2 ) ⊇ fc (Mj+1
) = M2j ) M2j+2 .
Applying fc , we now see that
fc (M2j ) = fc (M2j+2 ).
Again, Lemma 4.5 yields that fc is bounded.
In case (2),
fc (Mj−1 ) ⊇ fc (Mj ) = Pj−2,a and fc (Mj−1 ) ⊇ Mj−1 .
Thus
fc (Mj−1 ) ⊇ Pj−2,a + Mj−1 = Mj−2 ⊇ Mj−1 implies fc (Mj−1 ) = fc (Mj−2 ).
Now we are in the same set up as our Lemma but two steps up. If fc (Mj−2 ) = Mj−2 we
are done by case (1) above. Otherwise, fc (Mj−2 ) = fc (Mj−1 ) = Pj−4,b , some b ∈ K. Now
Now P2j−8,2c = fc (Mj−1 )2 ⊆ fc (M2j−2 ) ⊆ fc (M2j−4 ) ⊆ fc (P2j−8,2c ) which implies
fc (M2j−2 ) = fc (M2j−4 ). Again fc is bounded by Lemma 4.5.
¤
10
JANET C. VASSILEV
Lemma 4.7. If fc is a semiprime operation on K[[t2 , t3 ]] and fc (Mj ) = fc (Pj−2,b ) for some
j ≥ 4 and b ∈ K, then fc is a bounded semiprime operation.
Proof: As in the proof of Lemma 4.6 fc (Mj ) = fc (Pj−2,b ), implies that fc (Mj−1 ) =
fc (Mj−2 ). We now conclude by Lemma 4.6 that fc is also bounded.
¤
The following theorem describes the unbounded semiprime operations over K[[t2 , t3 ]].
Theorem 4.8. Let S be a nonempty subset of K, T any subset. If fc is an unbounded
semiprime operation over K[[t2 , t3 ]], then fc is either the identity or
(
I if I ⊇ Pi,a , a ∈
/ S or I ⊇ Pi+1,b , b ∈
/T
int
fi,S,T (I)
.
I if I ⊆ Mi+1 , I = Pi,a , a ∈ S or I = Pi+1,b , b ∈ T
Proof: Suppose fc is an unbounded semiprime operation over K[[t2 , t3 ]] which is not the
identity. Then fc (I) 6= I for some ideal I. If I = Mj for some j ≥ 2, then by Lemmas 4.5,
4.6 and 4.7, fc would be bounded, contradicting the unbounded assumption. Thus I must
be a principal ideal. If fc (Pk,a ) = fc (Pk+2,b ) for some k, then
fc (Pk,a ) = fc (Mk+2 ) = fc (Pk+2,b )
and Lemma 4.7 implies that fc is bounded contradicting the unboundedness assumption.
If fc (Pk,a ) = fc (Mk−1 ) for some k, then
fc (Pk,a ) = fc (Mk ) = fc (Mk−1 )
which is bounded by Lemma 4.6. Thus fc (Pk,a ) = Mk for all k = j or k ≥ j + 2. Note for
all b ∈ K,
fc (Pj+1,b ) ⊆ fc (Mj+1 ) = Mj+1 ,
int
thus fc (Pj+1,b ) = Pj+1,b or fc (Pj+1,b ) = Mj+1 . Hence, fc = fj,S,T
where S = {a ∈
K|fc (Pj,a ) = Mj } and T = {b ∈ K|fc (Pj+1,b ) = Mj+1 }.
¤
The bounded semiprime operations are given by the following theorem:
Theorem 4.9. The only bounded semiprime operations on K[[t2 , t3 ]] are of the form


I for I ⊇ Pm,a





Mm−1 for I = Pm−1,b , ∀b ∈ K
f
fm,a (I) = Mm for I = Pm,b , b 6= a, I = Pm+1,c , ∀c ∈ K or I = Mm+1


Pm,a for nonzero I ⊆ Pm,a



(0) if I = (0)

I for I ⊇ Pm,a



M
m−1 for I = Pm−1,b , ∀b ∈ K
f
gm,a
(I) =

Mm for I = Pm,b , b 6= a, I = Pm+1,c , ∀c ∈ K or I = Mm+1



Pm,a for I ⊆ Pm,a
for m ≥ 2 and a ∈ K,
STRUCTURE ON THE SET OF CLOSURE OPERATIONS OF A COMMUTATIVE RING
11

I for I ⊇ Pn,a , a ∈
/ S or I ⊇ Pn+1,b , b ∈
/T



I for P
m−1,c ⊆ I ⊆ J, J = Pn,a , a ∈ S or J = Pn+1,b , b ∈ T
f
fn,S,T,m
(I) =

Mm for nonzero I ⊆ Mm



(0) if I = (0)


/ S or I ⊇ Pn+1,b , b ∈
/T
I for I ⊇ Pn,a , a ∈
f
gn,S,T,m (I) = I for Pm−1,c ⊆ I ⊆ J, J = Pn,a , a ∈ S or J = Pn+1,b , b ∈ T


Mm for I ⊆ Mm
for m − 1 ≥ n ≥ 2, S 6= ∅ and if m = n + 1, T = K.
Proof: If fc is a bounded semiprime operation, then for small nonzero I, either
(1) fc (I) = Pm,a for I ⊆ Pm,a some a ∈ K or
(2) fc (I) = Mm for I ⊆ Mm
for some m ≥ 2.
Case (1): If fc (I) = Pm,a for I ⊆ Pm,a , then for fc to be semiprime, fc (Pi,b ) = Pi,b for
2 ≤ i ≤ m − 2 and all b ∈ K as the only factors of Pm,a are Pi,b 2 ≤ i ≤ m − 2 any b ∈ K.
Note that
Mm+2 ⊆ Pm+1,b ⊆ Mm+1 ⊆ Pm−1,c ,
b, c ∈ K but
Mm+3 ⊆ Mm+2 ⊆ Pm,a .
Thus fc (Mm+2 ) = fc (Mm+3 ) = Pm,a . However, Mm+1 ⊇ Pm+1,b are both incomparable
with Pm,a . But combining the above containments, we see that Pm,a ⊆ fc (Mm+1 ) ⊆
fc (Pm+1,b ). Thus
Mm ⊆ fc (Pm+1,b ) ⊆ fc (Mm+1 ).
Now P2,0 Pm+1,b = Pm+3,b so
P2,0 fc (Pm+1,b ) = fc (P2,0 )fc (Pm+1,b ) ⊆ fc (Pm+3,b ) = Pm,a .
Thus
fc (Pm+1,b ) ⊆ fc (Mm+1 ) ⊆ Pm−2,b .
Combining the above inclusions we see that fc (Pm+1,b ) = fc (Mm+1 ) = Pm−2,b or fc (Pm+1,b ) =
fc (Mm+1 ) = Mm .
Now if fc (Pm+1,b ) = Pm−2,b and as Pm+1,b ⊆ Pm−1,b then Pm−2,b ⊆ fc (Pm−1,b . However,
Pm−1,b and Pm−2,b are incomparable so
fc (Mm−2 ) ⊆ fc (Mm−1 ) = fc (Pm−1,b ).
Then
Pm−2,b = fc (Mm+1 ) ⊇ fc (P2,0 )fc (Mm−1 ) ⊇ P2,0 Mm−2 = Mm−1
but these two ideals are incomparable. So
fc (Pm+1,b ) = fc (Mm+1 ) = Mm .
This implies that
fc (Pm−1,b ) = Mm−1 .
12
JANET C. VASSILEV
f
f
Thus fc is fm,a
or gm,a
.
Case (2): If fc (I) = Mm for some I ⊆ Mm and m ≥ 2. Only the Pm−1,b is not comparable
to Mm . So clearly, as
Mm+1 ⊆ Pm−1,b and fc (Mm+1 ) = Mm
then fc (Pm−1,b ) = fc (Mm−1 ). It may be that fc (Pn,b ) = Mn for 2 ≤ n ≤ m − 1. Assume n
is the smallest 2 ≤ n ≤ m − 1 satisfying this property. Then for n + 2 ≤ k ≤ m − 1,
fc (Pk,b ) = Mk as Pk,b = Pj,a Pk−j,c , a + c = b
for 2 ≤ j ≤ k − n and
fc (Pk,b ) ⊇ fc (Pj,a )fc (Pk−j,c ) = Mk .
f
f
If fc (Pn+1,b ) = Mn+1 for all b ∈ K, then fc = fn,S,K,m
or fc = gn,S,K,m
depending on where
f
f
fc maps (0). If fc (Pn+1,b ) = (Pn+1,b ), for some b ∈ K then fc = fn,S,T,m
or fc = gn,S,T,m
depending on where fc maps (0).
¤
Combining the results of Theorems 4.8 and 4.9 and looking at compositions of the maps
obtained in the theorems we see the following:
Theorem 4.10. Let R = K[[t2 , t3 ]]. Then SR is the union of the monoids
f
int
f
M0 = {e} ∪ {fn,S,T
, fn,a
, fn,S,T,m
}
and
f
f
Mf = {e} ∪ {gn,a
, gn,S,T,m
}
where Mf is a left M0 -act but not a right M0 -act under composition.
Proof: First we look at all compositions of semiprime operations in M0 . Throughout,
we will denote K\{a} = aC . The compositions are as follows:

int

fm,S,T
if m + 2 ≤ n



int


fm,S,T ∪U if m + 1 = n
int
int
int
int
int
(M1) fm,S,T ◦ fn,U,V = fn,U,V ◦ fm,S,T = fm,S∪U,T
∪V if m = n


int
f

n,U,V ∪S if n + 1 = m


f int if n + 2 ≤ m
n,U,V
f

fn,S,T,m
if n + 1 ≤ m < l



f


fn,S,T ∪U,m if n + 1 = l ≤ m
f
f
f
int
int
(M2) fn,S,T,m ◦ fl,U,V = fl,U,V ◦ fn,S,T,m = fn,S∪U,T
∪V,m if n = l ≤ m − 1


f
f

l,U,S∪V,m if l + 1 = n ≤ m − 1


f f
l,U,V,m if l + 1 < n ≤ m − 1
 f
fm,a if m < l, m = l, a ∈
/ U or l = m − 1, a ∈
/V



f f
f
m−1,K,K,m if m = l, a ∈ U
int = f int ◦ f f
(M3) fm,a
◦ fl,U,V
m,a =
f
l,U,V

fm−1,U,K,m
if l = m − 1, a ∈ V


 f
fl,U,V,m if l < m − 1
STRUCTURE ON THE SET OF CLOSURE OPERATIONS OF A COMMUTATIVE RING
f
f
f
f
◦ fl,U,V,k
= fl,U,V,k
◦ fn,S,T,m
(M4) fn,S,T,m
f
f
f
f
(M5) fn,S,T,m
◦ fl,a
= fl,a
◦ fn,S,T,m
f
f
f
f
(M6) fn,a
◦ fm,b
= fm,b
◦ fn,a
13
 f
fn,S,T,m if n + 1 < m, l




f

fn,S,T

∪U,m if n + 1 = l ≤ m < k


f


fn,S,T ∪U,k if n + 1 = l ≤ k − 1 ≤ m − 1



f f
n,S∪U,T ∪V,m if n = l < m − 1 < k − 1
=
f

fn,S∪U,T

∪V,k if n = l < k − 1 ≤ m − 1


f


fl,U,S∪V,m if l + 1 = n ≤ m − 1 < k − 1



f


fn,U,S∪V,k
if l + 1 = n < k ≤ m


 f
fl,U,V,k if l + 1 < m, k
 f

fn,S,T,m if m ≤ l



f


fn,S,T,l if n + 1 < l ≤ m
f
= fn,K,T ∪aC ,l if n + 1 = l ≤ m


f

fn−1,K,K,n if n = l ≤ m − 1


 f
fl,a if l < n
 f
fn,a if n + 1 < m


 f

fn−1,K,K,n if m ≤ n ≤ m + 1
=
f

fm−1,K,K,m
if n + 1 = m


 f
fm,b if m + 1 < n
Clearly, M0 is a monoid and similar compositions show that Mf is a monoid. To see that
Mf is a left M0 -act but not a right M0 -act we look at the mixed compositions.
f
int = f int ◦ g f
(L1) gn,S,T,m
◦ fl,U,V
l,U,V
n,S,T,m
f
int = f int ◦ g f
(L2) gm,a
◦ fl,U,V
m,a
l,U,V
f
f
(L3) (a) gn,S,T,m
◦ fl,U,V,k
 f

gn,S,T,m if n + 1 ≤ m, l


 f


gn,S,T ∪U,m if n + 1 = l ≤ m
f
= gn,S∪U,T
∪V,m if n = l ≤ m − 1


f
g

l,U,S∪V,m if l + 1 = n ≤ m − 1


g f
l,U,V,m if l < n ≤ m − 1
 f
gm,a if m < l, m = l, a ∈
/ U or l = m − 1, a ∈
/V



g f
m−1,K,K,m if m = l, a ∈ U
=
f

gm−1,U,K,m
if l = m − 1, a ∈ V


 f
gl,U,V,m if l < m − 1
 f
gn,S,T,m if n + 1 < l, m ≤ k



g f
n,S,T ∪U,m if n + 1 = l < m ≤ k
=
f

gn,S∪U,T

∪V,m if n = l ≤ m − 1 ≤ k − 1


not a semiprime operation if k < m
14
JANET C. VASSILEV
f
f
(b) fl,U,V,k
◦ gn,S,T,m
(
 f
gn,S,T,m if n + 1 < l, m ≤ k




f

gn,S,T

∪U,m if n + 1 = l < m ≤ k


f


gn,S,T ∪U,k if n + 1 = l ≤ k < m



g f
n,S∪U,T ∪V,m if n = l ≤ m − 1 ≤ k − 1
=
f

gn,S∪U,T

∪V,k if n = l < k ≤ m


f


gl,U,V ∪S,m if l + 1 = n < m ≤ k



f


gl,U,V

∪S,k if l + 1 = n ≤ k < m

 f
gl,U,V,k if l + 1 < n, k ≤ m
f
gn,a
if n + 1 < m
not a semiprime operation if m ≤ n + 1
 f
gn,a if n + 1 < m



g f
f
f
n−1,K,K,n if m ≤ n ≤ m + 1
(b) fm,b
◦ gn,a
=
f

if n + 1 = m
gm−1,K,K,m


 f
gm,b if m + 1 < n
(
f
gn,S,T,m
if m ≤ l
f
f
(L5) (a) gn,S,T,m
◦ fl,a
=
not a semiprime operation if l < m
 f

gn,S,T,m if m ≤ l



f


gn,S,T,l if n + 1 < l ≤ m
f
f
f
(b) fl,a
◦ gn,S,T,m
= gn,K,T ∪aC ,l if n + 1 = l ≤ m


f

gn−1,K,K,n
if n = l ≤ m − 1



 f
g if l < n
( l,a
f
gl,a
if l < n
f
f
(L6) (a) gl,a ◦ fn,S,T,m =
not a semiprime operation if l ≥ n
 f

g
if m ≤ l

 n,S,T,m

f


gn,S,T,l if n + 1 < l ≤ m
f
f
f
(b) fn,S,T,m ◦ gl,a = gn,K,T ∪aC ,l if n + 1 = l ≤ m


f

gn−1,K,K,n
if n = l ≤ m − 1



 f
gl,a if l < n
(L4) (a)
f
gn,a
◦
f
fm,b
=
Hence Mf is a left M0 -act but not a right M0 -act.
¤
int
We will now see that the integral closure, f2,K,K , is the only prime operation besides the
identity.
int
Theorem 4.11. Let R = K[[t2 , t3 ]]. Then PR = {e, f2,K,K
}.
Proof: Suppose fc one of the other semiprime operations. Then for some i + 2 = j,
fc (Pi,a ) = Pi,a and fc (Pj,b ) = Mj for all b. Now Pj,a = t2 fc (Pj,a ) 6= fc (Pj,a ) = Mj . Hence
int
fc cannot be prime. Thus PR = {e, f2,K,K
}.
¤
STRUCTURE ON THE SET OF CLOSURE OPERATIONS OF A COMMUTATIVE RING
15
References
[Gi]
[HRR]
[HH]
[Na]
[Ra]
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271-396.
Hochster, M., Huneke, C., Tight closure, invariant theory, and the Brianon-Skoda theorem, J.
Amer. Math. Soc. 3 (1990), no. 1, 31–116.
¯
Nagy, A., Special Classes of Semigroups, Kluwer Academic Publishers, Dordrecht, 2001.
Ratliff, L., ∆-closures of ideals and rings, Trans. Amer. Math. Soc., 313 (1989), no. 1, 221–247.
The University of California, Riverside, CA 92521
E-mail address: [email protected]