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In this paper, we thoroughly study the Nakayama property and some related concepts. Also, we describe multiplication modules that, among other things, satisfy the Nakayama property. Next, we show that a ring 푅 is a Max ring if and only if all modules that can be generated by a finite or countable set have the weak Nakayama property. We prove that a ring 푅 is a perfect ring if and only if every module that can be generated by a finite or countable set has the Nakayama property. Finally, we present some categorical results on the aforementioned properties. [ABSTRACT FROM AUTHOR]

In this paper, we investigate when symbolic and ordinary powers of the parity binomial edge ideal of a graph fail to be equal. It turns out that if I G is the parity binomial edge ideal of a graph G, then in each of the following cases the symbolic power I G (t) and the ordinary power I G t are not equal for some t: (i) the clique number of G is greater than 3; (ii) G has a net; or (iii) G has a PT as an induced subgraph. [ABSTRACT FROM AUTHOR]

In this paper, we study Noetherian local rings R having a finite number of trace ideals. We proved that such rings are of dimension at most two. Furthermore, if the integral closure of R/H, where H is the zeroth local cohomology, is equi-dimensional, then the dimension of R is at most one. In the one-dimensional case, we can reduce to the situation that rings are Cohen–Macaulay. Then, we give a necessary condition to have a finite number of trace ideals in terms of the value set obtained by the canonical module. We also gave the correspondence between trace ideals of R and those of the endomorphism algebra of the maximal ideal of R when R has minimal multiplicity. [ABSTRACT FROM AUTHOR]

In this paper we examine some natural ideal conditions and show how graphs can be defined that give a visualization of these conditions. We examine the interplay between the multiplicative ideal theory and the graph theoretic structure of the associated graph. Behavior of these graphs under standard ring extensions are studied, and in conjunction with the theory, some classical results and connections are made. [ABSTRACT FROM AUTHOR]

A restricted dth power of an ideal I is obtained by restricting the exponent vectors allowed to appear on the "natural" generating set of Id, for some integer d. In this paper, we study homological properties of restricted powers of complete intersections. We construct a generalization of the L-complex construction of Buchsbaum and Eisenbud. We use this resolution to compute an explicit basis for the Koszul homology which allows us to deduce that the quotient defined by any restricted dth power of a complete intersection is Golod, and construct an algebra structure on this complex. [ABSTRACT FROM AUTHOR]

If I is a monomial ideal with linear quotients, then it has componentwise linear quotients. However, the converse of this statement is an open question. In this paper, we provide two classes of ideals for which the converse of this statement holds. First class is the componentwise polymatroidal ideals in K[x, y] and the second one is the componentwise polymatroidal ideals with strong exchange property. [ABSTRACT FROM AUTHOR]

In this paper, we investigate whether the symbolic and ordinary powers of a binomial edge ideal J G are equal. We show that the equality J G t = J G (t) holds for every t ≥ 1 when | Ass (J G) | = 2 . Moreover, if G is a caterpillar tree, then one has the same equality. Finally, we characterize the generalized caterpillar graphs which the equality of symbolic and ordinary powers of J G occurs. [ABSTRACT FROM AUTHOR]

In this paper, we prove the Grothendieck's Vanishing Theorem for finitely generated ideals over almost Dedekind domains and show that there exists a non-finitely generated prime ideal over a non-Noetherian almost Dedekind domain that does not satisfy the Grothendieck's Vanishing Theorem. Among the other results, by considering the attached prime ideals and associated prime ideals of representable modules over almost Dedekind domains, we show that if M is a representable R-module and for every non-zero submodule N of M, the ring R / Ann R (N) has acc on d-annihilators, then Att (Γ P (M)) ⊆ { 0 , P } . Also, if M is representable and for every non-zero submodule N of E (M) / M , the ring R / Ann R (N) has acc on d-annihilators, then Att (H P 1 (M)) ⊆ { 0 } , where E (M) is an injective envelope of M. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the structures of the cell modules of the complexified Green algebra R(D(H4)), where D(H4) is the Drinfel'd quantum double of Sweedler's 4-dimensional Hopf algebra H4. We show that R(D(H4)) has one infinite dimensional cell module, one 4-dimensional cell module generated by all finite dimensional indecomposable projective modules of D(H4) and infinitely many 2-dimensional cell modules. More precisely, we obtain the decompositions of all finite dimensional cell modules into the direct sum of indecomposable submodules, and show that the infinite dimensional cell module can be written as the direct sum of two infinite dimensional indecomposable submodules. [ABSTRACT FROM AUTHOR]

In this paper, we will study the higher order z-ideals of some special rings. We will characterize the z-ideals of the ring A [ [ X ] ] . Furthermore, we will give a similar result for the ring of idealization of an R-module M, R (+) M . [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce the concept of 2-absorbing coprimary submodule of a module as a generalization of strongly 2-absorbing second submodules and investigate some properties of this submodule class. We study the structure of 2-absorbing coprimary submodules of comultiplication modules, fully coidempotent modules and modules over a Noetherian ring. We also investigate some interrelations with 2-absorbing coprimary submodules, strongly 2-absorbing second submodules and strongly 2-absorbing secondary submodules. [ABSTRACT FROM AUTHOR]

A graded ideal I in K [ x 1 , ... , x n ] , where K is a field, is said to have almost maximal finite index if its minimal free resolution is linear up to the homological degree pd (I) - 2 , while it is not linear at the homological degree pd (I) - 1 , where pd (I) denotes the projective dimension of I. In this paper, we classify the graphs whose edge ideals have this property. This in particular shows that for edge ideals the property of having almost maximal finite index does not depend on the characteristic of K . We also compute the nonlinear Betti numbers of these ideals. Finally, we show that for the edge ideal I of a graph G with almost maximal finite index, the ideal I s has a linear resolution for s ≥ 2 if and only if the complementary graph G ¯ does not contain induced cycles of length 4. [ABSTRACT FROM AUTHOR]

Let R = K [ x 1 , … , x n ] be the polynomial ring in n variables over a field K and I be a monomial ideal generated in degree d. Bandari and Herzog (Eur. J. Combin.34 (2013) 752–763) conjectured that a monomial ideal I is polymatroidal if and only if all its monomial localizations have a linear resolution. In this paper, we give an affirmative answer to the conjecture in the following cases: (i) height (I) = n - 1 ; (ii) I contains at least n - 3 pure powers of the variables x 1 d , … , x n - 3 d ; (iii) I is a monomial ideal in at most four variables. [ABSTRACT FROM AUTHOR]

Let M be a module over a commutative ring R. In this paper, we continue our study about the Zariski topology-graph G (τ T) which was introduced in Ansari-Toroghy et al. (Commun Algebra 42:3283–3296, 2014). For a non-empty subset T of Spec (M) , we obtain useful characterizations for those modules M for which G (τ T) is a bipartite graph. Also, we prove that if G (τ T) is a tree, then G (τ T) is a star graph. Moreover, we study coloring of Zariski topology-graphs and investigate the interplay between χ (G (τ T)) and ω (G (τ T)) . [ABSTRACT FROM AUTHOR]

A well-known theorem of Kaplansky states that any projective module is a direct sum of countably generated modules. In this paper, we prove the w-version of this theorem, where w is a hereditary torsion theory for modules over a commutative ring. Communicated by Silvana Bazzoni [ABSTRACT FROM AUTHOR]

Let R a commutative ring with identity and M be a unitary R-module. In this paper, we investigate some properties of n-absorbing submodules of M as a generalization of 2-absorbing submodules. We also define the classical n-absorbing submodule, a proper submodule N of an R-module M is called a classical n-absorbing submodule if whenever a 1 a 2 ... a n + 1 m ∈ N for a 1 , a 2 , ... , a n + 1 ∈ R and m ∈ M , there are n of a i 's whose product with m is in N. Furthermore, we give some characterizations of n-absorbing and classical n-absorbing submodules under some conditions. [ABSTRACT FROM AUTHOR]

In this paper we generalize a result, concerning a depth equality over local rings, proved independently by Araya and Yoshino, and Iyengar. Our result exploits complexity, a concept which was initially defined by Alperin for finitely generated modules over group algebras, introduced and studied in local algebra by Avramov, and subsequently further developed by Bergh. [ABSTRACT FROM AUTHOR]

This paper contributes several results to the analytic theory of ideals. The main new concept is that of a prereduction (together with the closely related type-prereduction): if R is a ring and A ⊂ I (proper subset) are proper ideals in R, then A is called a prereduction of I in case A is not a reduction of I, but each ideal between A and I is a reduction of I. It is shown that each non-nilpotent proper ideal in a Noetherian ring has at least one prereduction, and a number of the basic properties of the ideals in the set P (I) of all prereductions of I are proved. Also, if I is a non-nilpotent proper ideal in a local (Noetherian) ring (R, M), then: There is a natural one-to-one correspondence between the set E (I) of the equivalence classes of the type-prereductions of I and the set Q (I) of the maximal relevant ideals in the Rees ring of I (that is, the homogeneous prime ideals Q in the ring R [ t − 1 , I t ] (t an indeterminate) such that I t ⊈ Q and the Krull dimension of R [ t − 1 , I t ] / Q is equal to one). P (I n) is the set of maximal elements in { Q [ n ] | Q ∈ Q (I) } for all positive integers n, and, for each Q ∈ Q (I) , Q [ m ] ∈ P (I m) for all large integers m. A complete description is given of the prereductions and the elements in E (I) for each ideal I that is generated by analytically independent elements. If R/M is algebraically closed and I is a non-nilpotent and non-principal ideal in R, then there is a natural one-to-one correspondence between the sets P (I) and E (I) , and P (I n) = { Q [ n ] | Q ∈ Q (I) } for all positive integer n. [ABSTRACT FROM AUTHOR]

Let R = k [ x 1 , ... , x n ] be the polynomial ring in n variables over a field k and let I be a matroidal ideal of degree d. In this paper, we study the unmixedness properties and the arithmetical rank of I. Moreover, we show that a r a (I) = n - d + 1 . This answers the conjecture made by Chiang-Hsieh (Comm Algebra 38:944–952, 2010, Conjecture). [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to generalize a result of T.G. Lucas which states that if R is a reduced commutative ring and M is a flat R-module, then the idealization R ⋉ M is an A -ring if and only if R is an A -ring [14, Proposition 3.5]. In effect, we drop the reduceness hypotheses and prove that, given an arbitrary commutative ring R and any submodule M of a flat R-module F, R ⋉ M is an A -ring (resp., S A -ring) if and only if R is an A -ring (resp., S A -ring). [ABSTRACT FROM AUTHOR]

In this paper, we prove that if R is a commutative ring with unity and M is a finitely generated R-module, then M is Noetherian if and only if for every prime ideal P of R with A n n (M) ⊆ P , there exists a finitely generated submodule N P of M such that P M ⊆ N P ⊆ M (P) . [ABSTRACT FROM AUTHOR]

In this paper, we introduce the dual notion of fully cancellation modules (that is fully cocancellation modules) over a commutative ring and investigate some basic properties of this class of modules. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the class of von Neumann regular modules over commutative rings. More precisely, we introduce a characterization of regular modules, and then, we study some properties of these modules in viewpoint of this characterization. Among other things, we show that the Nakayama's Lemma and Krull's intersection theorem hold for this class of modules. Also, some explicit expressions for submodules of regular modules are introduced. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and Specs(M) denote the set all second submodules of an R-module M. In this paper, we investigate various properties of Specs(M) with respect to different topologies. We investigate the dual Zariski topology from the point of view of separation axioms, spectral spaces and combinatorial dimension. We establish conditions for Specs(M) to be a spectral space with respect to quasi-Zariski topology and second classical Zariski topology. We also present some conditions under which a module is cotop. [ABSTRACT FROM AUTHOR]

In this paper we will prove some properties of the algebras $W(I)$, $\;c^*(I)$ and $c(I)$ which arise when considering the notion of $I$-convergence of real sequences. By $I$-convergence of a real sequence we mean the convergence according to the filter associated to an ideal $I$ on the set $\mathbb{N}$. $W(I)$ is the set of sequences having finite $I$-variation, $ c^*$ and $c$ stand for two kinds of $I$-convergence. In particular, we are studying the maximal ideals of $W(I)$ in the cases where $I$ is admissible or non-admissible. We do that by using the topological properties of the semi-normed space $l^{\infty}(I)$. Finally, we discuss the question whether all maximal ideals $P$ of $W(I)$ are of the form $P=c_0(\mathcal{M})\cap W(I)$ where $\mathcal{M}$ is a maximal ideal of $2^{\mathbb{N}}$. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the structures of the cell modules of the complexified Green algebra R(D(H4)), where D(H4) is the Drinfel'd quantum double of Sweedler's 4-dimensional Hopf algebra H4. We show that R(D(H4)) has one infinite dimensional cell module, one 4-dimensional cell module generated by all finite dimensional indecomposable projective modules of D(H4) and infinitely many 2-dimensional cell modules. More precisely, we obtain the decompositions of all finite dimensional cell modules into the direct sum of indecomposable submodules, and show that the infinite dimensional cell module can be written as the direct sum of two infinite dimensional indecomposable submodules. [ABSTRACT FROM AUTHOR]

LetKbe a field andR = K[x1,…,xn] be the polynomial ring in the variablesx1,…,xn. In this paper we prove that when 𝔄 = {𝔭1,…, 𝔭m} andare two arbitrary sets of monomial prime ideals ofR, then there exist monomial idealsIandJofRsuch thatI ⊆ J, Ass∞(I) = 𝔄 ∪ 𝔅, AssR(R/J) = 𝔅, and AssR(J/I) = 𝔄 \ 𝔅, where Ass∞(I) is the stable set of associated prime ideals ofI. Also we show that when 𝔭1,…, 𝔭mare nonzero monomial prime ideals ofRgenerated by disjoint nonempty subsets of {x1,…,xn}, then there exists a square-free monomial idealIsuch that AssR(R/Ik) = Ass∞(I) = {𝔭1,…, 𝔭m} for allk ≥ 1. [ABSTRACT FROM PUBLISHER]