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On the stable set of associated prime ideals of
monomial ideals and square-free monomial
ideals
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Kazem Khashyarmanesh and Mehrdad Nasernejad
The 10th Seminar on Commutative Algebra and Related Topics,
18-19 December 2013
(In honor of Professor Hossein Zakeri)
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Cover ideals
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Let R be a commutative Noetherian ring and I be an ideal of R.
Brodmann showed that Ass(R/I s ) = Ass(R/I s+1 ) for all
.sufficiently large s.
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A natural question arises in the context of Brodmann’s
Theorem:
(∗) Is it true that
AssR (R/I) ⊆ AssR (R/I 2 ) ⊆ · · · ⊆ AssR (R/I k ) ⊆ · · · ?
McAdama presented an example which says, in general, the
above question has negative answer.
a
McAdam, Asymptotic prime divisors, Lecture Notes in Mathematics 103,
Springer-Verlag,
New York, 1983.
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Cover ideals
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Let R be a commutative Noetherian ring and I be an ideal of R.
Brodmann showed that Ass(R/I s ) = Ass(R/I s+1 ) for all
.sufficiently large s.
.
A natural question arises in the context of Brodmann’s
Theorem:
(∗) Is it true that
AssR (R/I) ⊆ AssR (R/I 2 ) ⊆ · · · ⊆ AssR (R/I k ) ⊆ · · · ?
McAdama presented an example which says, in general, the
above question has negative answer.
a
McAdam, Asymptotic prime divisors, Lecture Notes in Mathematics 103,
Springer-Verlag,
New York, 1983.
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Cover ideals
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The ideal I is said to have the persistence property if
s ) ⊆ Ass(R/I s+1 ) for all s ≥ 1.
Ass(R/I
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Let k be a fixed field and R = k [x1 , . . . , xn ] a polynomial ring
over k . An ideal in R is monomial if it is generated by a set of
monomials. A monomial ideal is square-free if it has a
generating set of monomials, where the exponent of each
variable is at most 1.
Problem : Do all square-free monomial ideals have the
persistence
property?
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Cover ideals
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The ideal I is said to have the persistence property if
s ) ⊆ Ass(R/I s+1 ) for all s ≥ 1.
Ass(R/I
.
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Let k be a fixed field and R = k [x1 , . . . , xn ] a polynomial ring
over k . An ideal in R is monomial if it is generated by a set of
monomials. A monomial ideal is square-free if it has a
generating set of monomials, where the exponent of each
variable is at most 1.
Problem : Do all square-free monomial ideals have the
persistence
property?
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Cover ideals
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We recall the following definitions and construction.
A graph G is said to be critically s-chromatic if χ(G) = s but
χ(G\x) = s − 1 for every x ∈ V (G), where G\x denotes the
graph obtained from G by removing the vertex x and all edges
incident to x. A graph that is critically s-chromatic for some s is
.called critical.
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For any vertex xi ∈ V (G), the expansion of G at the vertex xi is
the graph G′ = G[{xi }] whose vertex set is given by
V (G′ ) = (V (G)\{xi }) ∪ {xi,1 , xi,2 } and whose edge set has form
E(G′ ) = {{u, v } ∈ E(G)|u ̸= xi and v ̸= xi }
∪{{u,
xi,1 }, {u, xi,2 }|{u, xi } ∈ E(G)} ∪ {{xi,1 , xi,2 }}.
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Cover ideals
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We recall the following definitions and construction.
A graph G is said to be critically s-chromatic if χ(G) = s but
χ(G\x) = s − 1 for every x ∈ V (G), where G\x denotes the
graph obtained from G by removing the vertex x and all edges
incident to x. A graph that is critically s-chromatic for some s is
.called critical.
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For any vertex xi ∈ V (G), the expansion of G at the vertex xi is
the graph G′ = G[{xi }] whose vertex set is given by
V (G′ ) = (V (G)\{xi }) ∪ {xi,1 , xi,2 } and whose edge set has form
E(G′ ) = {{u, v } ∈ E(G)|u ̸= xi and v ̸= xi }
∪{{u,
xi,1 }, {u, xi,2 }|{u, xi } ∈ E(G)} ∪ {{xi,1 , xi,2 }}.
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Cover ideals
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Equivalently, G[{xi }] is formed by replacing the vertex xi with
the clique K2 on the vertex set {xi,1 , xi,2 }. For any W ⊆ V (G),
the expansion of G at W , denoted G[W ], is formed by
successively
expanding all the vertices of W (in any order).
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C. A. Francisco, H. T. Há and A. Van Tuyl
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a Conjecture . Let s be a positive integer, and let G be a finite
simple graph that is critically s-chromatic. Then there exists a
subset W ⊆ V (G) such that G[W ] is a critically
(s + 1)-chromatic graph.
a
C. A. Francisco, H. T. Há and A. Van Tuyl, A conjecture on critical graphs
and connections to the persistence of associated primes, Discrete Math. 310
(2010),
2176-2182.
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Cover ideals
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Equivalently, G[{xi }] is formed by replacing the vertex xi with
the clique K2 on the vertex set {xi,1 , xi,2 }. For any W ⊆ V (G),
the expansion of G at W , denoted G[W ], is formed by
successively
expanding all the vertices of W (in any order).
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C. A. Francisco, H. T. Há and A. Van Tuyl
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a Conjecture . Let s be a positive integer, and let G be a finite
simple graph that is critically s-chromatic. Then there exists a
subset W ⊆ V (G) such that G[W ] is a critically
(s + 1)-chromatic graph.
a
C. A. Francisco, H. T. Há and A. Van Tuyl, A conjecture on critical graphs
and connections to the persistence of associated primes, Discrete Math. 310
(2010),
2176-2182.
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Cover ideals
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Definition
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Let G be a finite simple graph on the vertex set
V (G) = {x1 ,∩
. . . , xn }. The cover ideal of G is the monomial ideal
J = J(G) = {xi ,xj }∈E(G) (xi , xj ) ⊆ R = k [x1 , . . . , xn ].
..
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It is not hard to see that
J(G)
= (xi1 . . . xir | W = {xi1 , . . . , xir } is a minimal vertex cover of G).
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Cover ideals
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Definition
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Let G be a finite simple graph on the vertex set
V (G) = {x1 ,∩
. . . , xn }. The cover ideal of G is the monomial ideal
J = J(G) = {xi ,xj }∈E(G) (xi , xj ) ⊆ R = k [x1 , . . . , xn ].
..
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It is not hard to see that
J(G)
= (xi1 . . . xir | W = {xi1 , . . . , xir } is a minimal vertex cover of G).
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Cover ideals
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(Francisco et al.(2010))
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Theorem. Let G be a finite simple graph with cover ideal
J = J(G). Let s ⩾ 1 and assume that the conjecture holds for
(s + 1). Then
Ass(R/J s ) ⊆ Ass(R/J s+1 ).
In particular, if the conjecture holds for all s, then J has the
.persistence property.
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Cover ideals
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Theorem
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(T. Kaiser, M. Stehlik and R. Skrekovski)a The cover ideal of the
following graph does not have the persistence property.
a
Replication in critical graphs and the persistence of monomial ideals, J.
Combin.
Theory, Ser. A, (to appear).
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Cover ideals
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Finally, Morey and Villarreala prove persistence for edge ideals
I of any graphs containing a leaf (a vertex of degree 1).
a
S. Morey and R. H. Villarreal, Edge ideals: algebraic and combinatorial
properties, Progress in Commutative Algebra, Combinatorics and Homology,
Vol.
. 1, 2012, 85-126.
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persistence property
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Definition
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The smallest integer k0 such integerAss(R/I k ) = Ass(R/I k+1 )
for all k ≥ k0 , denoted astab(I), is called the index of stability for
the associated prime ideals of I. Also the set AssR (R/I k0 ) is
called the stable set of associated prime ideals of I, which is
denoted
by Ass∞ (I).
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persistence property
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It has been shown, see McAdama and Bandari, Herzog and
Hibi,b that given any numbern there exists an ideal I in a
suitable graded ring R and a prime ideal p of R such that, for all
k ≤ n, p ∈ Ass(R/I k ) if k is even and p ̸∈ Ass(R/I k ) if k is odd.
a
S McAdam, Asymptotic prime divisors, Lecture Notes in Mathematics
103, Springer-Verlag, New York, 1983.
b
S. Bandari, J. Herzog, T. Hibi, Monomial ideals whose depth function has
any
. given number of strict local maxima, Preprint 2011
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persistence property
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Definition
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The edge ideal of a simple graph G, denoted by I(G), is the
ideal of R generated by all square-free monomials xi xj such
that {xi , xj } ∈ E(G). The assignment G −→ I(G) gives a natural
one to one correspondence between the family of graphs and
the family of monomial ideals generated by square-free
..monomials of degree 2.
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persistence property
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Theorem
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Martinez-Bernal, Morey and Villarreala Let G be a graph and let
I = I(G) be its edge ideal. Then
Ass(R/I k ) ⊆ Ass(R/I k+1 )
for all k .
a
J. Martinez-Bernal, S. Morey and R. H. Villarreal, Associated primes of
powers
of edge ideals, Collect. Math. 63 (2012), 361-374.
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persistence property
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Definition
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J. Herzog, A. Qureshia , Let p ∈ V (I). We say that I satisfies the
strong persistence property with respect to p if, for all k and all
f ∈ (Ipk : pRp )\Ipk , there exists g ∈ Ip such that fg ̸∈ Ipk +1 . The
ideal I is said to satisfy the strong persistence property if it
satisfies the strong persistence property for all p ∈ V (I).
a
J. Herzog, A. Qureshi, Persistence and stability properties of powers of
ideals
(2012)
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Theorem
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The ideal I of R satisfies the strong persistence property if and
only
if I k+1 : I = I k for all k.
..
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persistence property
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Definition
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J. Herzog, A. Qureshia , Let p ∈ V (I). We say that I satisfies the
strong persistence property with respect to p if, for all k and all
f ∈ (Ipk : pRp )\Ipk , there exists g ∈ Ip such that fg ̸∈ Ipk +1 . The
ideal I is said to satisfy the strong persistence property if it
satisfies the strong persistence property for all p ∈ V (I).
a
J. Herzog, A. Qureshi, Persistence and stability properties of powers of
ideals
(2012)
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Theorem
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The ideal I of R satisfies the strong persistence property if and
only
if I k+1 : I = I k for all k.
..
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persistence property
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Definition
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An ideal I is polymatroidal if the following “exchange condition”
is satisfied: For monomials u = x1a1 . . . xnan and v = x1b1 . . . xnbn
belonging to G(I) and, for each i with ai > bi , one has j with
a
.. j < bj such that xj u/xi ∈ G(I).
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Proposition
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Let I be a polymatroidal ideal. Then I satisfies the strong persistence
property.
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persistence property
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Definition
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An ideal I is polymatroidal if the following “exchange condition”
is satisfied: For monomials u = x1a1 . . . xnan and v = x1b1 . . . xnbn
belonging to G(I) and, for each i with ai > bi , one has j with
a
.. j < bj such that xj u/xi ∈ G(I).
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Proposition
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Let I be a polymatroidal ideal. Then I satisfies the strong persistence
property.
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persistence property
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Definition
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A graph G = (V (G), E(G)) is perfect if, for every induced
subgraph
GS , with S ⊆ V (G), we have χ(GS ) = ω(GS ).
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persistence property
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Theorem
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Francisco, Há and Van Tuyla Let G be a perfect graph with
cover ideal J. Then
(1) Ass(R/J s ) ⊆ Ass(R/J s+1 ) for all integers s ≥ 1.
(2)
χ(G)−1
∞
∪
∪
s
Ass(R/J ) =
Ass(R/J s ).
s=1
s=1
a
C. A. Francisco, H. T. Há, and A. Van Tuyl, Colorings of hypergraphs,
perfect graphs, and associated primes of powers of monomial ideals, J.
Algebra
331 (2011), 224-242.
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persistence property
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Lemma
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I be a monomial ideal. Then Ass(I t−1 /I t ) = Ass(R/I t ).
.Let
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Stable set
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Theorem
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Hoaa Let
√
√
B = max{d(rs + s + d)( r )r +1 ( 2d)(r +1)(s−1)
, s(s + r )4 sr +2 d 2 (2d 2 )s
2 −s+1
}.
Then we have
Ass(I n /I n+1 ) = Ass(I B /I B+1 )
for all n ≥ B.
a
L.T. Hoa, Stability of associated primes of monomial ideals, Vietnam J.
Math.
34 (2006), no. 4, 473-487.
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Stable set
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Example
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Let d ≥ 4 and
I = (x d , x d−1 y , xy d−1 , y d , x 2 y d−2 z) ⊂ K [x, y , z].
Then
Ass(I n−1 /I n ) = {(x, y , z), (x, y )} if n < d − 2, and
n−1 /I n ) = {(x, y )} if n ≥ d − 2.
Ass(I
..
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Stable set
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Theorem
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Bayati, Herzog and Rinaldoa Let p1 , ..., pm ⊆ R be an arbitrary
collection of nonzero monomial prime ideals. Then there exists
a monomial ideal I of R such that Ass∞ (I) = {p1 , ..., pm }.
a
Sh. Bayati, J. Herzog and G. Rinaldo, On the stable set of associated
prime
ideals of a monomial ideal, Arch. Math. 98, No. 3, 213-217 (2012).
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Stable set
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Question
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suppose we are given two sets A = {p1 , ..., pℓ } and
B = {p′1 , . . . , p′m } of monomial prime ideals such that the
minimal elements of these sets with respect to inclusion are the
same. For which such sets does exist a monomial ideal I such
that
Ass(R/I) = A and Ass∞ (I) = B?
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For example, there is no monomial ideal I with
Ass(R/I)
= {(x1 ), (x2 )} and Ass∞ (I) = {(x1 ), (x2 ), (x1 , x2 )}.
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Results
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Remark
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Let p1 , . . . , pm be non-zero monomial prime ideals of R such
that |G(pi )| ≤ |G(pj )| for all 1 ≤ i < j ≤ m. Then, for all d ∈ N,
m−1 d
4d
2
AssR (R/pd1 ∩ p2d
2 ∩ p3 ∩ · · · ∩ pm
) = {p1 , . . . , pm }.
This means that there exist infinite monomial ideals with
..associated prime {p1 , . . . , pm }.
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Results
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Theorem
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Let A = {p1 , . . . , pm } and B = {p′1 , . . . , p′t } be two arbitrary sets
of monomial prime ideals of R. Then there exist monomial
ideals I and J of R with the following properties:
(i) AssR (R/I) = A ∪ B, AssR (R/J) = B and
..
(ii) I ⊆ J, AssR (J/I) = A\B.
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Results
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Theorem
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Let A = {p1 , . . . , pm } and B = {p′1 , . . . , p′t } be two arbitrary sets
of monomial prime ideals of R. Then there exist monomial
ideals I and J of R such that
(i) Ass∞ (I) = A ∪ B, AssR (R/J) = B and
..
(ii) I ⊆ J, AssR (J/I) = A\B.
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Results
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Theorem
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Let A = {p1 , . . . , pm } be a set of non-zero monomial prime
ideals of R such that they are generated by disjoint non-empty
subsets of {x1 , . . . , xn }. Also, suppose that {A1 , . . . , Ar } is a
partition of A. Then there exist square-free monomial ideals
I1 , . . . , Ir such that, for all positive integers k1 , . . . , kr , d,
(i) AssR (R/Iiki ) = Ai ,
(ii) AssR (R/I1k1 d . . . Irkr d ) = {p1 , . . . , pm }, and
(iii) Ass∞ (I1k1 . . . Irkr ) = Ass∞ (I1k1 ) ∪ · · · ∪ Ass∞ (Irkr ).
..
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Results
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Suppose that I is a monomial ideal of R with minimal generating
set {u1 , . . . , um }. We say that I satisfies the condition (♯) if there
exists a nonnegative integer i with 1 ≤ i ≤ m such that
i−1 c
αm
i+1
(u1α1 . . . ui−1
uiαi ui+1
. . . um
uj :R ui ) =
α
α
i−1 c
αm
i+1
. . . um
(uj :R ui )
u1α1 . . . ui−1
uiαi ui+1
α
α
αi
for all j = 1, . . . , m with j ̸= i and α1 , . . . , αm ≥ 0, where uc
i
means
that this term is omitted.
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Results
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Theorem
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Every ideal satisfies the condition (♯) has the persistence
..property.
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Results
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Definition
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Let I be a monomial ideal of R with the unique minimal set of
monomial generators G(I) = {u1 , . . . , um }. Then we say that I
is a weakly monomial ideal if there exists i ∈ N with 1 ≤ i ≤ m
such that each monomial uj has no common factor with ui for
all
.. j ∈ N with 1 ≤ j ≤ m and j ̸= i.
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Example
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Consider the ideal I = (x32 x5 x63 , x13 x22 x44 , x16 x23 x74 , x22 x74 x45 ) in the
polynomial ring R = K [x1 , x2 , x3 , x4 , x5 , x6 , x7 ]. It is easy to see
..that I is a weakly monomial ideal of R.
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Results
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Definition
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Let I be a monomial ideal of R with the unique minimal set of
monomial generators G(I) = {u1 , . . . , um }. Then we say that I
is a weakly monomial ideal if there exists i ∈ N with 1 ≤ i ≤ m
such that each monomial uj has no common factor with ui for
all
.. j ∈ N with 1 ≤ j ≤ m and j ̸= i.
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Example
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Consider the ideal I = (x32 x5 x63 , x13 x22 x44 , x16 x23 x74 , x22 x74 x45 ) in the
polynomial ring R = K [x1 , x2 , x3 , x4 , x5 , x6 , x7 ]. It is easy to see
..that I is a weakly monomial ideal of R.
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Results
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Definition
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Let I be a monomial ideal of R with the unique minimal set of
monomial generators G(I) = {u1 , . . . , um }. Then we say that I
is a strongly monomial ideal if there exist i ∈ N with 1 ≤ i ≤ m
and monomials g and w in R such that ui = wg, gcd(w, g) = 1,
and
for all j ∈ N with 1 ≤ j ̸= i ≤ m, gcd(uj , ui ) = w.
..
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Results
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Example
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Consider the ideal
I = (x1 x2 x33 x45 , x12 x23 x34 x53 x65 , x12 x2 x35 x52 x6 , x13 x22 x36 x64 )
in the polynomial ring R = K [x1 , x2 , x3 , x4 , x5 , x6 ]. Then, by
setting
u1 := x1 x2 x33 x45 ,
u2 := x12 x23 x34 x53 x65 ,
u3 := x12 x2 x35 x52 x6 ,
u4 := x13 x22 x36 x64 ,
i := 1 and w := x1 x2 x33 , clearly that I is a strongly monomial
ideal of R.
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Results
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Theorem
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Every strongly (or weakly) monomial ideal of R satisfies
condition
(♯).
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Thanks For Your Patience
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