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Let B be a local (Noetherian) ring and suppose that B has n associated prime ideals where n ≥ 2 . We identify sufficient conditions for there to exist a local (Noetherian) subring S of B such that S and B have the same completion and S has exactly n − 1 associated prime ideals. We include applications and consequences of this result. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the Krull dimension-dependent elements of a Noetherian commutative ring. Let x , y be non-unit elements of a commutative ring R. x , y are called Krull dimension-dependent elements, whenever dim R / (R x + R y) = min { dim R / R x , dim R / R y }. We investigate the elements of a ring according to this property. Among the many results, we characterize the rings that all elements of them are Krull dimension-dependent and we call them, closed under the Krull dimension. Moreover, we determine the structure of the rings with Krull dimension at most 1, that are closed under the Krull dimension. [ABSTRACT FROM AUTHOR]

The authors of A. Benhissi and S. Hizem [Nonnil-Noetherian rings and SFT property, Rocky Mountain J. Math. 41 (2011) 1483–1500] proved that the primary decomposition of a nonnil-ideal of a nonnil-Noetherian ring always exists. We will show in this paper that this primary decomposition is minimal. We next prove that over any nonnil-Noetherian ring, a sub-class of modules admit a primary decomposition. In the second part of this paper, we give some properties related to ϕ-chained and ϕ-Prüfer rings. [ABSTRACT FROM AUTHOR]

In this paper, we study the cofiniteness of generalized local cohomology modules with respect to an ideal of a commutative Noetherian ring and the class of modules in dimension less than a fixed integer. As consequences, we present some results on the ordinary cofiniteness, weakly cofiniteness, and finiteness of the set of associated prime ideals of generalized local cohomology modules. [ABSTRACT FROM AUTHOR]

In this paper, our main focus is to explore different classes of nearly normally torsion-free ideals. We first characterize all finite simple connected graphs with nearly normally torsion-free cover ideals. Next, we characterize all normally torsion-free t-spread principal Borel ideals that can also be viewed as edge ideals of uniform multipartite hypergraphs. [ABSTRACT FROM AUTHOR]

Introduction Throughout this paper, is a commutative Noetherian ring with non-zero identity, is an ideal of , is a finitely generated -module, ‎and is an arbitrary -module which is not necessarily finitely generated. Let L be a finitely generated R-module. Grothendieck, in [11], conjectured that is finitely generated for all . In [12], ‎Hartshorne gave a counter-example and raised the question whether is finitely generated for all and . The th generalized local cohomology module of and with respect to , was introduced by Herzog in [14]. It is clear that is just the ordinary local cohomology module of with respect to . As a generalization of Hartshorne's question, we have the following question for generalized local cohomology modules (see [25, Question 2.7]). Question. When is finitely generated for all and ? In this paper, we study in general for a finitely generated -module and an arbitrary -module . Material and methods The main tool used in the proofs of the main results of this paper is the spectral sequences. Results and discussion We present some technical results (Lemma 2.1 and Theorems 2.2, 2.9, and 2.14) which show that, in certain situation, for non-negative integers , , , and with , and the -modules and are in a Serre subcategory of the category of -modules (i.e. the class of -modules which is closed under taking submodules, quotients, and extensions). Conclusion We apply the main results of this paper to some Serre subcategories (e.g. the class of zero -modules and the class of finitely generated -modules) and deduce some properties of generalized local cohomology modules. In Corollaries 3.1-3.3, we present some upper bounds for the injective dimension and the Bass numbers of generalized local cohomology modules. We study cofiniteness and minimaxness of generalized local cohomology modules in Corollaries 3.4-3.8. Recall that, an -module is said to be -cofinite (resp. minimax) if and is finitely generated for all [12] (resp. there is a finitely generated submodule of such that is Artinian [26]) where . We show that if is finitely generated for all and is minimax for all , then is -cofinite for all and is finitely generated (Corollary 3.5). We prove that if is finitely generated for all , where is the arithmetic rank of , and is -cofinite for all , then is also an -cofinite -module (Corollary 3.6). We show that if is local, , and is finitely generated for all , then is -cofinite for all if and only if is finitely generated for all (Corollary 3.7). We also prove that if is local, , is finitely generated for all , and (or ) is -cofinite for all , then is -cofinite for all (Corollary 3.8). In Corollary 3.9, we state the weakest possible conditions which yield the finiteness of associated prime ideals of generalized local cohomology modules. Note that, one can apply the main results of this paper to other Serre subcategories to deduce more properties of generalized local cohomology modules../files/site1/files/71/15.pdf

Introduction Throughout this paper, is a commutative ring with non-zero identity and is an -module. The study of injective modules is very important in commutative algebra and homological Algebra. Any product of (even finitely many) injective modules is injective; conversly, if a direct product of modules is injective, then each module is injective. Every direct sum of finitely many injective modules is injective. In general, submodules, factor modules, or infinite direct sum of injective modules need not be injective. Every submodule of every injective module is injective if and only if the ring is Artinian semisimple. Also every factor module of every injective module is injective if and only if the ring is Hereditary. Finally every infinite direct sum of injective modules is injective if and only if the ring is Noetherian. In this paper we study some new propertis of this modules. Material and methods The main tool used in the proofs of the main results of this paper is the properties of injective modules and injective envelopes. Results and discussion We present some new properties of injective envelopes, injective modules, prime submodules and maximal submodules. Conclusion We prove the following results: Over finitely generated multiplication modules, every prime submodule is irreducible. If N is a prime submodule of finitely generated multiplication R-module M such that E(M/N) is finitely generated, then N is a maximal submodule of M. Also we give several corollaries for this note. Also we find relations beetwen Artinian ring, Noetherian ring, indecomposable injective modules and injective cogenerators of modules../files/site1/files/71/4.pdf

Let n and t be non-negative integers, R a commutative Noetherian ring with dim (R) ≤ n + 2 , a an ideal of R, M and N finite R-modules, and X an arbitrary R-module. We prove that if Ext R i (M / a M , X) is an FD < n R-module for all i ≤ t , then H a i (M , X) is an ( FD < n , a) -cofinite R-module for all i < t, Hom R (R / a , H a t (M , X)) is an FD < n R-module, and { p ∈ Ass R (H a i (M , X)) : dim (R / p) ≥ n } is a finite set for all i ≤ t. It shows that H a i (M , N) is ( FD < n , a) -cofinite and { p ∈ Ass R (H a i (M , N)) : dim (R / p) ≥ n } is finite for all i. In particular, H a i (M , N) is a -cofinite for all i whenever dim (R) ≤ 2. Also, Ass R (H a i (M , N)) is finite for all i when R is semi-local with dim (R) ≤ 3. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring and M be an R -module. The compressed essential graph of M , denoted by E G (M) is a simple undirected graph associated to M whose vertices are classes of torsion elements of M and two distinct classes [ m ] and [ m ′ ] are adjacent if and only if Ann R (m) + Ann R (m ′) is an essential ideal of R. In this paper, we study diameter and girth of E G (M) and we characterize all modules for which the compressed essential graph is connected. Moreover, it is proved that ω (E G (M)) = | Ass R (M) | , whenever R is Noetherian and M is a finitely generated multiplication module with r ( Ann R (M)) = 0. [ABSTRACT FROM AUTHOR]

Suppose that is a commutative Noetherian ring with identity, , are ideals of , and let be a finitely generated -module. Let be the th local cohomology functor with respect to . In this paper, we show that the -module is always finitely generated. Moreover, we provide sufficient conditions such that the modules are finitely generated. [ABSTRACT FROM AUTHOR]

Let M and N be two finitely generated graded modules over a standard graded Noetherian ring R = ⊕ n≥0 Rn. In this paper we show that if R0 is semi-local of dimension ≤ 2 then, the set AssR0 ( HiR+ (M, N)n ) is asymptotically stable for n → −∞ in some special cases. Also, we study the torsion-freeness of graded generalized local cohomology modules HiR+ (M, N). Finally, the tame loci T i (M, N) of (M, N) will be considered and some sufficient conditions are proposed for the openness of these sets in the Zariski topology. [ABSTRACT FROM AUTHOR]

Let (R , 𝔪) be a Noetherian local ring, I , J two ideals of R , and M , N two finitely generated R -modules. It is first shown that Ass R (𝔉 I dim   M − 1 (M)) is a finite set. We also prove that except the maximal ideal 𝔪 , the set Ass R (𝔉 I d − 1 (N n)) is stable for large n , where we use N n to denote R -module J n M / J n + 1 M or M / J n M and d is the eventual value of dim   N n . [ABSTRACT FROM AUTHOR]

The notion of generalized Laskerian modules was introduced recently by the authors. In this paper, we continue our investigation of such modules. Moreover, we are going to generalize some results on finiteness of associated primes of local cohomology modules to the category of generalized Laskerian modules. [ABSTRACT FROM AUTHOR]

Let ℐ be a set of ideals of a commutative Noetherian ring R. We use the notion of ℐ -closure operation which is a semiprime closure operation on submodules of modules to introduce the class of ℐ -Laskerian modules. This enables us to investigate the set of associated prime ideals of certain ℐ -closed submodules of local cohomology modules. [ABSTRACT FROM AUTHOR]

Let I denote an ideal of a Noetherian ring R and N a nonzero finitely generated R -module. It is shown that if the 𝔭 -symbolic topology is equivalent to the 𝔭 -adic topology on N , for all 𝔭 ∈ mAss R N / I N , then the I -symbolic topology on N is equivalent to the I -adic topology on N. Moreover, we show that if Ass R 𝔭 ̂ N 𝔭 ̂ consists of a single prime ideal, for all 𝔭 ∈ A ∗ (I , N) , then the I -adic and the I -symbolic topologies on N are equivalent. Finally, it is shown that if for every one-dimensional prime ideal 𝔭 in Supp (N) , the 𝔭 -adic and the 𝔭 -symbolic topologies are equivalent on N , then N is unmixed and Ass R N has only one element. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring with non-zero identity, an ideal of R, M and X finite R–modules, and t a non-negative integer such that is –cofinite for all i < t. In this article, we show that and are finite, and is finite if and only if is finite. As a consequence, we observe that is a finite set. [ABSTRACT FROM AUTHOR]

Let S be a polynomial ring over a field of characteristic zero, and let I⊂S be an ideal of intersection type assumed moreover to have no embedded primary component. Our main goal in this paper is to provide an effective sufficient condition for a given monomial prime ideal to avoid the sets of prime divisors of the powers of I, and in particular to avoid the celebrated set of asymptotic prime divisors of I, which will follow from a new and quite surprising double-colon stability property. Further, we briefly describe some other applications, e.g., on the topology of a suitable blowing-up. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a semiprime closure operation on submodules of a module. Several properties of this closure operation are studied and some applications are provided. [ABSTRACT FROM AUTHOR]

In this paper, by using elementary tools of commutative algebra,we prove the persistence property for two especial classes of rings. In fact, thispaper has two main sections. In the first main section, we let R be a Dedekindring and I be a proper ideal of R. We prove that if I1, . . . , In are non-zeroproper ideals of R, then Ass1(Ik11 . . . Iknn ) = Ass1(Ik11 ) [ · · · [ Ass1(Iknn )for all k1, . . . , kn ≥ 1, where for an ideal J of R, Ass1(J) is the stable setof associated primes of J. Moreover, we prove that every non-zero ideal ina Dedekind ring is Ratliff-Rush closed, normally torsion-free and also has astrongly superficial element. Especially, we show that if R = R(R, I) is theRees ring of R with respect to I, as a subring of R[t, u] with u = t−1, then uRhas no irrelevant prime divisor.In the second main section, we prove that every non-zero finitely generatedideal in a Pr¨ufer domain has the persistence property with respect to weaklyassociated prime ideals. Finally, we extend the notion of persistence propertyof ideals to the persistence property for rings.

Let be a field and be a polynomial ring in the variables . In this paper, we introduce two classes of monomial ideals of , which have the following properties: Next, we characterize these monomial ideals. In the sequel, we give some combinatorial aspects. We conclude this paper with constructing new monomial ideals, which have the persistence property. [ABSTRACT FROM AUTHOR]

Let I and J be two ideals of a commutative Noetherian ring R and M be an R-module. For a non-negative integer n it is shown that, if the sets AssR(ExtnR(R/I,M)) and SuppR(ExtiR(R/I,HjI,J(M))) are finite for all i ≤ n + 1 and all j < n, then so is AssR(HomR(R/I,HnI,J(M))). We also study the finiteness of AssR(ExtiR(R/I,HnI,J(M))) for i = 1, 2. [ABSTRACT FROM AUTHOR]

Let R = K[x1, . . ., xn] be the polynomial ring, where K is a field and x1, . . ., xn are indeterminates. Let ... = {p1, . . ., pm} and B = {p′1, . . ., p′t} be two arbitrary sets of monomial prime ideals of R such that the elements of A ∪B are minimal with respect to inclusion. We show that there exist unique square-free monomial ideals I and J of R with the following properties: (i) AssR(R/I) = A, AssR(R/J) = B; and, (ii) I ⊆ J and AssR(J/I) = A\B. In the sequel, we investigate the associated prime ideals of monomial ideals under squeezing. Finally, we state two recursive formulas for associated prime ideals. [ABSTRACT FROM AUTHOR]

The associated prime ideals of powers of polymatroidal ideals are studied, including the stable set of associated prime ideals of this class of ideals. It is shown that polymatroidal ideals have the persistence property and for transversal polymatroids and polymatroidal ideals of Veronese type the index of stability and the stable set of associated ideals is determined explicitly. [ABSTRACT FROM AUTHOR]

It is shown that any set of nonzero monomial prime ideals can be realized as the stable set of associated prime ideals of a monomial ideal. Moreover, an algorithm is given to compute the stable set of associated prime ideals of a monomial ideal. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring with non-zero identity, a an ideal of R and M,N two R-modules. The main purpose of this paper is to study the circumstances under which, for fixed integers j ϵ N0 and n ϵ N, the R-modules Extj R(N,Hna (M)) and TorRj (N,Hna (M)) are weakly Laskerian or Matlis reflexive. In this way, we also get to some results about the associated primes, coassociated primes and Bass numbers of Hna (M). [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring, 픞 an ideal of R, and M an R-module. The purpose of this paper is to show that if M is finitely generated and dim M/픞M > 1, then the R-module ∪{N|N is a submodule of $H^{t}_{\mathfrak a}(M)$ and dim N ≤ 1} is 픞-cominimax and for some x ∈ R is Rx + 픞-cofinite, where t ≔ gdepth(픞, M). For any nonnegative integer l, it is also shown that if R is semi-local and M is weakly Laskerian, then for any submodule N of $H_{\mathfrak a}^{l}(M)$ with dim N ≤ 1 the associated primes of $H_{\mathfrak a}^{l}(M)/N$ are finite, whenever ${\rm dim} H_{\mathfrak a}^{i}(M)\leq 1$ for all i < l. Finally, we show that if (R, 픪) is local, M is finitely generated, ${\rm dim} H_{\mathfrak a}^{i}(M)\leq 1$ for all i < l, and ${\rm dim} H_{\mathfrak a}^l(M) > 1$ then there exists a generalized regular sequence x1, ..., xl ∈ 픞 on M such that ${\rm Ass}_R H_{\mathfrak a}^{l}(M)\backslash \{\mathfrak m\}= {\rm Ass}_R M/(x_1,\ldots, x_l)M \backslash \{\mathfrak m\}$. [ABSTRACT FROM AUTHOR]

Let 픞 be an ideal of a commutative Noetherian local ring R, and let M and N be two finitely generated R-modules. Let t be a positive integer. It is shown that if the support of the generalized local cohomology module $H_{\frak a}^i(M,N)$ is finite for all i < t, then the set of associated prime ideals of the generalized local cohomology module $H_{\frak a}^t(M,N)$ is finite. Also, if the support of the local cohomology module $H_{\frak a}^i(N)$ is finite for all i < t, then the set $\big(\bigcup_{i\in {\Bbb N}}{\rm Ass}_R ({\rm Ext}_R^t(M/{\frak a}^i M,N))\big) \cap\{{\frak p}\in {\rm Spec}(R): \dim R/{\frak p}>1\}$ is finite. Moreover, we prove that gdepth(픞+Ann(M),N) is the least integer t such that the support of the generalized local cohomology module $H_{\frak a}^t(M,N)$ is an infinite set. [ABSTRACT FROM AUTHOR]

Abstract: Let I be a monomial ideal of a polynomial ring and the maximal degree of minimal generators of I. In this paper, we explicitly determine a number in terms of r and such that for all . Furthermore, our is almost sharp. [Copyright &y& Elsevier]

A certain set of associated primes of the Matlis duality of any top local cohomology module of a complete filter local ring is characterized. Also, it is proved that the set of associated primes of a finitely generated module over a four-dimensional local ring is finite. [ABSTRACT FROM AUTHOR]

Let αa be an ideal of a commutative Noetizerian ring R with identity and let M and N be two finitely generated R-modules. Let t be a positive integer. It is shown that ASSR (Hαt (M, N)) is contained in the union of the sets ASSR (ExtRi(M, Hαt-i (N))), where 0 ≤ i ≤ t. As an immediate consequence, it follows that if either Hαi(N) is finitely generated for all i < t or SUPPR (Hαi(N)) is finite for all i < t, then ASSR (Hαt(M, N)) is finite. Also, we prove that if d = pd(M) and n = dim (N) are finite, then Hαd+n (M, N) is Artinian. In particular, ASSR (Hαd+n(M, N)) is afinite set consisting of maximal ideals. [ABSTRACT FROM AUTHOR]

The notion of weakly Laskerian modules was introduced recently by the authors. Let R be a commutative Noetherian ring with identity, ?? an ideal of R, and M a weakly Laskerian module. It is shown that if ?? is principal, then the set of associated primes of the local cohomology module is finite for all i = 0. We also prove that when R is local, then is finite for all i = 0 in the following cases: (1) dim R = 3, (2) dim R/?? = 1, (3) M is Cohen-Macaulay, and for any ideal ??, with l = grade(??, M), is weakly Laskerian. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity. For a finitely generated R -module M , the notion of associated prime submodules of M is defined. It is shown that this notion inherits most of the essential properties of the usual notion of associated prime ideals. In particular, it is proven that for a Noetherian multiplication module M , the set of associated prime submodules of M coincides with the set of M -radicals of primary submodules of M which appear in a minimal primary decomposition of the zero submodule of M . Also, Anderson's (1994) theorem is extended to minimal prime submodules in a certain type of modules. [ABSTRACT FROM AUTHOR]