REND. SEM MAT.
UNIVERS. POLITECN. TORINO
Vol. 47°, 2 (1989)
M.C. Vipera (*)
MONOMIAL IDEALS OF POLYNOMIALS
OVER NOETHERIAN RINGS
S u n t o . Si generalizzano alcuni risultati noti sugli ideali monomiali a coefficienti in
un campo agli ideali monomiali a coefficienti in un anello aritmetico. In particolare
si fornisce un algoritmo per la decomposizione primaria.
Introduction
Monomial ideals over a field have been thoroughly studied by R. Kummer and B. Renschuch in [4] and [5]. In those papers it is shown the existence of natural algorithms, which allow elementary effective computation
of intersections, primary decompositions, Hilbert functions, etc., far from the
computational complexity of the case of general ideals. In this paper, many
results given in [4] are extended to the case of more general rings of coefficients, First some basic facts are proved for arbitrary Noetherian rings of
coefficients. Among them, a characterization of primary and quasi-primary
monomial ideals. In the case of monomial ideals over a field, similar characterizations have been given in terms of their "minimum basis", which is
essentially unique. Clearly, in the case of a more general ring of coefficients,
monomial ideals may admit different irredundant bases. So we define a new
invariant for a monomial ideal, the so-called "canonical pseudobasis" (see § 1),
and we give characterizations in terms of it. Then we make the further assumption that the ring of coefficients is a ZPI-ring, i.e. we suppose that the
lattice of ideals is distributive. Under this assumption, and by using pseudo( ) Research partially supported by MPI.
126
bases, we generalize almost all the results given in [4]. In particular we give
a very simple algorithm for the primary decomposition of a monomial ideal
and characterize radical monomial ideals.
Now, let us fix some basic notation and terminology.
We always denote by A a commutative Noetherian ring with 1 ^ 0 ,
and by ft = A[Xi,Xz, ...,Xn] the ring of polynomials in n indeterminates
with coefficients in A. Polynomials are denoted by small letters, like / , g,
etc., monomials by capitals, like F , G.
Ideals of ft are also denoted by capitals (/, J, etc.), while ideals of
A are denoted by small gothic letters: a, b , etc.
If S is a set of polynomials (including the case S C A), (S) always
denotes the ideal of R generated by S. For the sake of brevity we put:
r
(a,G,
Gr) = (a) + £(G,) .
1= 1
Sums and intersections of ideals are always finite.
The radical of / is denoted by \/7, as usual.
An ideal J of .ft is said to be a monomial ideal li it is generated by the
set of the monomials belonging to I or, equivaleritly, if / has a monomial
basis, i.e. a (finite) basis formed by monomials.
The proofs of the following statements are an easy generalization of the
case of polynomials over a fiels (see [4]).
(I) Let / be an ideal of ft. The following are equivalent:
(a) / is a monomial ideal.
(b) If g G ft and g — Y^ G, is the reduced representation of g as a
sum of monomials, then g e I if and only if G» € /.Vt.
(II) If / and J are monomial ideals, then so are /-f J, IJ} In J and I: J.
If H, L are monic monomials, we will denote their least common multiple by [H,L]. If a,b are ideals of A, then (a) n (H) = (ai/), where
a// = {a//|a<=a}, and (atf) D (bL) = ((a n b)[H,L]).
127
1 - Preliminary facts. B a s e s and pseudobases
The proof of the following proposition is left to the reader.
PROPOSITION l . l . (a) Let I be a monomial ideal, I = ^ ( a * / , , ) , and let
G = bH be a monomial (a,,6 € A; Li and H
Gel
if and only if b € V ] Aah.
monic monomials).
Then
Lh\H
(b) Let I — ]|P(a;L;), J = E;(bjGj),
where a< 's and bj's are ideals
i
of A and the Li's, Gj's
are monic monomials,
then J Cl
if and only if,
LH\GJ
DEFINITION. Let / be a monomial ideal. By pseudobasis of I we mean a
finite set of pairs {(a,, L,)}, where each a; is an ideal of A, a< ^ {0}, the
Lj's are distinct monic monomials and / = Y^(a,Li).
i
A pseudobasis {(a;,L;)} is said to be minimal if it does not properly
contain any pseudobasis, or, equivalently, if
a, ^
^
ak
Vj .
Lh\L,
A minimal pseudobasis can be easily obtained in the following way: let
B be an irredundant monomial basis of / ,
B = {anLi, ... ,ai,« l /'i, ... ,ar>iLr, ...,artSrLr}
,
with Li a monic monomial V,, and X, Y £* for i ^ *•• Put a, = Y^^4atJ c A.
It is easy to see that {(aj,Lj),'} is a minimal pseudobasis of /.
The following theorem shows that the monic monomials of a minimal
pseudobasis of / are uniquely determined.
THEOREM 1.2. / / / is monomial ideal and {(a,,!,)}[_!, {0Jj^j)}j=i
^re
minimal pseudobases of I, then r = s and, for each i, Li = Gj for some
h
Proof. Suppose, for some 'i, Li ^ Gj for all j . From Prop. 1.1(b) we
have a, c ])P b/, and b/, C V^ a* for all h. But Lk\Gh implies Ljb|£»,
"Gh\Li
Lk\Gh
128
with Lk ^ Li, whence a, C Y^ a,. This contradicts the minimality of the
L,\Li
pseudobasis {(a»,£j)}.
Similarly, every Gj equals some L,, therefore r = s.
•
Let / be a monomial ideal of R, and G a monic monomial. Denote
CI(G)
= {aeA\aG€l}
= (I: (G)) D A .
Obviously, if G | //, then c/(G) C c/(//). If {(a,-,L)} is a pseudobasis
of / , then for every monic monomial H £ R one has c/(//) = Y^ a/».
COROLLARY 1.3. For every monomial ideal I there exists a minimal pseudobasis {(hi, Li)} such that, for each minimal pseudobasis {(a,, L,)} of I,
one has a, C b,- for every i.
Proof. Clearly, I = ^ ( a ^ L , )
implies / = ^ ( c / ( L , ) L t ) , hence, if {(a;,£*)}
i
is a minimal pseudobasis for
»i C cj(Li) for all i. •
»'
/,
then so is {(c/(Z/,), L,)}. Clearly one has
Obviously, the property in the statement of Cor. 1.3 uniquely determines the pseudobasis {(<:/(£,),£,)}. We shall call it the canonical pseudobasis
of / and denote it by CP(I).
Suppose every c/(Z,-) which appears in CP(I) is a principal ideal
Vi, C[(Li) = <LiA. Then the basis {<*;£<}, is called the canonical basis of •/.
It is an irredundant basis and its cardinality is minimal. We can deduce from
Thm. 1.2 and Cor. 1.3 that, if {&;//>} is another irredundant basis, then, for
every j , Hj = L, for some i and 6; G a,A. If A is a principal ideal domain,
then every monomial ideal has a unique (up to associated elements) canonical
basis.
Example. Let / = (36, 12A', 9A', 7V, lbXY) C Z[X,V]. The given basis is
not irredundant. In fact / = (36, 12.Y, 9A\ 1Y) = (36, 3Ar, 7Y) = (36, 3Ar, Y).
These three bases are all irredundant, the second and third one have minimal
cardinality, and only the last one is the canonical basis.
129
2 - P r i m e and primary monomial ideals
As before, let
ideal in R = A[XUX2,
Put a 0 = c/(l)
CP(I). If a 0 = {0},
Thus, in any case we
A be a (Noetherian) ring, and let / be a monomial
...,*„].
= / H A If a 0 ^ {0}, then obviously the pair ( a 0 , l ) G
then we add the pair . ({0}., 1) to C P ( / ) , as (a 0 ,1).
shall write CP(I) = {(sii}Li)}ri=0 with L0 = 1.
THEOREM 2.1. Let I be a monomial ideal and {(a^L,-)}^ its canonical
pseudobasis. Then I is prime if and only if:
(i) a 0 is prime)
(ii) a, = A for all i > 0;
(iii) Li = A"/,, for all i > 0, where h, 1 < h < n.
Proof. If / satisfies the given conditions, that is / is of the form
(p,yYj,, ..., A' ir ), with p a prime ideal of A, then, obviously R/I
is
an integral domain.
Conversely, let / be prime. Then a 0 = / 0 A is also prime. Now
suppose a, ^ A for some i > 0. Clearly a; D a 0 , but a; ^ a 0 because of the
minimality of the pseudobasis. Let a € a ; - a 0 . Then aLi 6 / , which is prime,
and L{ £ I. Thus a e IDA = a 0 , which is a contradiction. This proves (ii).
Then the minimality of the pseudobasis yields Lj\Li, for i ^ j ^ 0, hence
(iii) easily follows •
We note that, in the above theorem (as in the others in this section), the
canonical pseudobasis cannot be replaced by a different minimal pseudobasis.
Recall that an ideal is called quasi-primary if its radical is a prime ideal.
THEOREM 2.2. Let I be a monomial ideal, and let CP(I) = {(a,-,!,-)}^Then I is quasi-primary if and only if
(i) a0 is quasi-primary;
(ii) if i > 0, then either a* C y/&o or there is h < n such that Xh \ Li
and (A,X%)€CP(I)
for some (3 > 0.
Furthermore, if I is quasi-primary, then \/7 = (N/ao,yY7l, ...,Xj t ), where the
Xjh's are the indeterminates for which (A, Xff)G CP(I) for some ph > 0.
Proof. Let / be quasi-primary and denote P = y/1 and p = y/s^. Then
p = VlnA = y/lnA = PDA which is prime.
130
Now assume a, £ p and let a G a,- - p . Since ah{ 6 / C P, U G P.
This implies Xh G P for some Xh which divides £,. Thus, some power of
Xh belongs to /. If (3 = min{7 G IN | X% G / } , then it is easy to see that
(A,xZ)eCP(I).
Conversely, let P = (p,A r ; i , ... ,X J t ), where p = ^/a^, and Xjx, ... ,Aj,
are the indeterminates such that X?* G / for some (3jh G IN (put P — (p)
if 8 = 0). Then I cPcVI.
*
THEOREM 2.3. Let I be a monomial ideal and let CP{I) — {(a,, £i)}!=oT/ien J is primary if and only if
^
(i) ao is primary;
(ii) if Xh | L,- Merc there exists (3 G IN swc/i </m< (4, Xf) G CP(7,J;
(iii) /or e^er^/ monic monomial II, either cj(II) = A or cf(H)/ is
primary, where p = >/ao.
p-
Proof. Let / be primary. Then (i) is trivial. On the other hand, if Xh \ L{,
write Li = X%G with Xh\G.
Let a G a, -
^
at. Then I BaLi = aGX%,
but aG £ / . Hence some powjer of Xh belongs to /. If X% is the one with
minimum exponent, then (A,X^) G CP(I).
Finally, let H be a monic monomial such that cj(H) ^ A and let
a G c 7 ( # ) . Then <z/7 G / , but # £ /. Hence a G ^ n \ / 7 = p . Thus ct(H) C p,
and so y/cj(H) = p , since a 0 C c/(//). Moreover, if a6 G c/(/f) and
b£c/(H),
then a(bH) £ I and fc# g/." Thus a G v l r i \ / 7 = p and so c / ( # )
is p-primary.
Conversely, from Thm. 2.2 we know that / quasi-primary with radical P = (p,A'i, ...,A 3 ) (up to a permutation of the indeterminates), where
X\, ...,XS are those (and only those) indeterminates such that (A,X^h) G
CP(I) for some j3h G IN. Now let f,g G R, with / g J and # £ P . We
want to show that fg 0 /. Without loss of generality, we may assume that
no monomial occurring in the representation of / belongs to /.
Let F be a monomial of / of minimum (total) degree with respect to
A'i,...,A' s and let L be the monic part of F which involves only
X\,...,XS.
Then we can write / = qL + f\, where q does not involve any A/, (h =-1, ..., s)
and /i and L are linearly independent over A[X8+i,.... ,Xn]. Also write
g .= g' -f g", where g" is the sum of the monomials in the representation
of g which are divisible by some Xh (1 < h < s). Since g $• P, it follows
131
g' # P. In particular g' £ 0. Thus fg = qg'L + fig' + fg", where, because
of the minimality of the degree of L, all the monomials occurring in qg'L
also occur in the representation of fg. Since no monomial occurring in qL
belongs to /, we have c/(£) / A. Thus, by (iii), c/(£) is p-primary, and
hence (c/(L)) is (p)-primary. Moreover q £ (c/(£)), g' £ (p) c P, therefore
qg' £ (cj(L)). This means that there is a monomial aG occurring in qg'
such that a £ cj(L). Now, because of (ii), the monomials of the canonical
pseudobasis involve only Xi,...,Xa,
while G involves only the remaining
variables, and so ci(GL) = c/(L). Thus aGL £ /, but aGL occurs in qg'L,
hence in fg, as we wished. •
REMARK. Condition (iii) in the above theorem requires to be checked only
in a finite number of cases. In fact, for every //, c/(//) = c/(//'), where W
is the least common multiple of those monomials occuring in the canonical
pseudobasis which divide H.
We note that, if A is a principal ideal domain, then condition (iii) of
thm. 2.3 can be replaced by:
(iii)'
. If ao = {0} then as = i4, Vt > 0 .
Therefore, in this case, every monomial ideal between a primary monomial ideal and its radical P, is also P-primary. In particular every sum of
P-primary monomial ideals is a P-primary ideal. Moreover, every product of
P-primary monomial ideals is P-primary.
Now we go back to the general case of any Noetherian ring A and give
a sufficient condition for a monomial ideal to be irreducible.
i
PROPOSITION 2.4. Let I = (a, A'f1, ...,X r °') 'with 0 < r < n, and a an
irreducible ideal of A. Then I is irreducible.
Proof. It follows from Thm. 2.3 that / is primary with radical P =
(p, Xi, ...,A' r ), where .p = v/a. We will prove that Is is irreducible in Rs
where S = R- P. By Thm. IV.34 of [10], this is equivalent to the condition
that (Is : Ps)/Is
is a principal ideal. Since / : P = 14 [(a : p ) J | x r _ 1 \,
it is enough to show that there exists 6 £ a : p such that, for every c 6 a : p ,
the image, c, of c in Rs/Js is a multiple of 6. Since a is an irreducible
ideal of A, then a8 : p 9 /a 9 is a principal ideal, where s = A — p. Since
s C £, tha conclusion easily follow. •
132
We end this section by giving a necessary condition for a monomial ideal
to be radical.
PROPOSITION 2.5. Let
{(a.,£.)},W
I
be a radical monomial
ideal and let GP(I) =
Then
(i) X\\Li
for all i and
h<n
(ii) For every monic monomial G G R, ci(G)
is radical..
Proof. If Li = X%H, then XhH € /, hence aXhH e I Va € a,-. But this is
clearly false if a G a, -
}]
a/,.
To prove (ii), notice that bkG € / implies bG € I.
•
3 - P r i m a r y decomposition under a "distributivity" assumption
Algorithms have been given for the primary decomposition of ideals of
polynomials over the ring of integers in [8], over a larger class of rings of
coefficients in [3]. As in the case of polynomials over a field, a much simpler
algorithm can be used for monomial ideals, provided the ring of coefficients is
an arithmetic ring.
We recall that a Notherian ring A is said to be arithmetic (or a ZPIring) if it satisfies one of the following equivalent conditions:
Dl)
a n ( b + c) = ( a n b ) + (aflc)
Va,b,c ideals of A *
D2)
a + (b D c) = (a + b) n (a + c)
Va, b, c ideals of A .
Other equivalent conditions can be found in [1], [2]. Notice that arithmetic integral domains are precisely Dedekind domains.
Throughout this section, except in the last proposition, A will denote
a ZPI-ring and R = A[X\,X2, ...,X n ] a polynomial ring over A.
Now we will use pseudobases again, in order to deduce from the above
distributive properties an algorithm for the primary decomposition of monomial ideals.
PROPOSITION 3.1. Let I = ^ ( a ^ L i ) , J = ]T](bj//j), be monomial ideals of
R,
where a;, b ; are ideals of A and Li, Hj are monic monomials.
Then
133
7 n J = £((a i nb J )[L,,7/ j ]).
».J
Proof. Let cG £ 10 J, with G monic; by 1.1(a) we have c € I y ^ a/» I fl
n{LHk\ahk)
=E%\2(*hnhk).
Thus cG G ^
Now L J G, and //* | G imply [Lh,Hk] | G.
((a^ n b*)[L/,,//*)), and this shows one inclusion. The other
Lh\G
one is obvious.
•
COROLLARY 3.2. The two distributive properties of intersection and sum
hold for monomial ideals of R.
DEFINITION. We will say that a monomial ideal 7 is an ideal of powers if
it is of the form (ao, A'"1, ...,A' s a '), 0 < 5 < », up to a permutation of the
indeterminates.
LEMMA 3.3. Every monomial ideal in ft is a finite intersection of ideals of
powers.
r
Proof
Let 7 = (a 0 ) + ]jP(aiL;). Let s be the number of "mixed" terms of
this sum, that is the number of terms (a»L;) (i > 0) for which either a, / A
or Li involves at least two indeterminates. We will prove that, if s > 0, then
7 can be expressed as an intersection of ideals such that the number of these
terms is less than s.
Let (ajLi) be mixed and let L{ = X?\ ...,X a '. Then
/ = («i)n (f| (*;;)J + (a0) + £>£,) =
= f (a, + a 0 ) + £ > £ , ) ) n If) UX%) + (ao) + £ > £ , )
by 3.2. The conclusion follows by induction.
•
Later on (thm. 3.8) we will use the following:
REMARK. In the proof of the above lemma we have expressed 7 = (a 0 ) +
134
r
^(atL,)
as an intersection of ideals b + X ^ / * " ) '
t'=l
wnere
b *s t n e
sum
°f
k
some aj's and every X?* divides L t , for some i.
We will show, now, that every monomial ideal in R can be expressed
as an (irredundant) intersection of irreducible monomial ideals.
PROPOSITION 3.4. A monomial ideal in R is irreducible if and only if it is
of the form (ao, X{*\ ... ,X?') where ao is irreducible.
Proof. Every irreducible monomial ideal must be an ideal of powers, by the
previous lemma. Moreover, a 0 must be irreducible, otherwise / would be
reducible by Cor. 3.2. See Prop. 2.4 for the converse. •
THEOREM 3.5. Every monomial ideal I
irreducible monomial ideals.
of R is a finite intersection of
Proof. Let / be an ideal of powers, / = (ao, A' J*1,. ... ,X?r),
and let ao = Q q j ,
i
l
where every <\j is an irreducible ideal of A. Then / = P|(qj, X° ,
.
by Cor. 3.2. The general case follows by Lemma 3.3.
...,X"r),
j
•
Lemma 3.3 and theorem 3.5 allow us to easily find a primary decomposition of a monomial ideal in R. In fact we write / as an intersection
of irreducible monomial ideals. Notice that there is an unique way to write
/ as an irredundant intersection of irreducible monomiarideals. The proof
is the same as that of Thin. 13(b) of [4]. A normal primary decomposition
can easily be found. By lemma 3.1, we can compute intersection of monomial
ideals over A by computing intersection of ideals of A.
COROLLARY 3.6. Every monomial ideal I of R has a normal primary
decomposition where all the components are monomial ideals. In particular,
isolated components and all associated prime ideals of I are monomial ideals.
Moreover radicals of monomial ideals are monomial.
We need a lemma in order to characterize radical monomial ideals.
LEMMA 3.7. Let a,b be radical ideals of (an aritmetic ring) A. Then a + b
135
is radical.
Proof. We can assume, without loss of generality, that A is reduced. Let
xk G a + b. By D l , xk G ( a n (**)) + ( b n (**)). Then xk = axk + 6**, with
axfc G a, 6a;* € b , whence ax G a and bx G b. If p is a prime ideal
of .4 such that x G p , then ar* = (a 4- b)xk implies (a-f 6) 4-p = 1-f p.
Therefore x + p = (a + b)x 4- p for every prime ideal p of A. This implies
x = (a 4-fc)z= as 4- 6# G a 4- b. •
THEOREM 3.8. Let I be a monomial ideal of R and let CP(I) = {(a*, £i)},r=oThen I is radical if and only if
(i) a; is radical V<;
(ii) X2h\Li V,- and V/i < n.
Proof. Suppose / satisfies (i), (ii). By Lemma 3.7 and by the Remark after
Lemma 3.3, J is an intersection of ideals of the form (b, X ; i , ...,Aj ( ), where
b is radical. By Cor. 3.2 and Thm. 2.1, (b, A';,, ...,A r Jf ) is an intersection
of prime ideals.
The converse has been already proved (Prop. 2.5).
COROLLARY 3.9.
Let
I
be a monomial ideal of
•
R, and let CP(I) =
r
{(a,,L,)}J_ 0 .
Then
VI = V^o + ^ ( v ^ G , ) ,
never Li; = X^1 ... X?'y
COROLLARY 3.10. / / /
I 4- J is a radical ideal.
where d = Xjl ...Xjt,
whe-
with cu- > 0 for every k.
and J are radical monomial ideals of R,
then
The following proposition allow us, in some particular cases, to recognize
radical (not monomial) ideals.
Let us endow the set of monic monomials in R with a term-ordering
which is compatible with the product. For / G R, denote by lt(f) the
leading term of / , with its coefficient, and, for an ideal / of R, put
Lt(i) = ({U(f) |
fei}).
PROPOSITION 3.11. Let A be any Noetherian ring and let I C R =
A[Xi,X2, .-.,>Vn] any ideal. If Lt(I) is a radical ideal, then I is radical.
136
Proof. First assume A is reduced. It easy to see that, for each / G R, and
for each k G IN, one has U(fk) = (lt{f))k. Therefore
Lt(I) c Lt(Vl) C y/Lt(I) ;
Now, if Lt(I) is radical, then one has Lt(I) = Lt(y/I) and, from [3],
Cor. 2.10, one gets \fl - I,
If A is not reduced, let n be the nilradical of A and <f> : R —+
A/ii[X\,X2, ...,Xn] the natural homomorfism. For any ideal J in R put
J = </»(J). Since Ker <fr = (n) is the nilradical of R, one has J = v^7 if and
only if (n) c J and J is radical.
^
Therefore Lt(I) radical implies Lt(I) radical and n C Lt(I)y so n C / ,
hence (n) C / .
We now have to show that 7 is radical. Since A/n is reduced, it is
sufficient to show that Lt(l) is radical. In fact, we show that Lt(I) = Lt(I).
Let all G Lt(I), where a € A, a = <f>(a) and / / is a monic monomial. There
must be an f£l
such that a'H is an effective monomial of / , with suitable
a' = a mod n, and such that each monomial of / "greater" than a'H has
a nilpotent coefficient. Then f = fx + / 2 , where fi G (n) and a'H is the
leading term of f2. From /i G / , it follows / 2 G / , so a'H G Lt(I) and
aH G Lt(I). The other inclusion is clear. •
Acknowledgement.
The author expresses her gratitude to Prof. S. Guazzone, who suggested her the theme of monomial ideals, and provided her with many valuable
advices.
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Maria Cristina VIPERA
Dipartimento di Matematica - Universita di Perugia
Via Vanvitelli 1 - 06100 PERUGIA
Pervenuto in redazione il 21.11.1988
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