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We provide criteria for identifying exact pairs of zero-divisors from zero-divisor graphs of commutative rings, and extend these criteria to compressed zero-divisor graphs. Finally, our results are translated as constructions for exact zero-divisor subgraphs. [ABSTRACT FROM AUTHOR]

We investigate the structure and properties of an Artinian monomial complete intersection quotient A (n , d) = [ x 1 , ... , x n ] / (x 1 d , ... , x n d). We construct explicit homogeneous bases of A (n , d) that are compatible with the S n -module structure for n = 3 , all exponents d ≥ 3 and all homogeneous degrees j ≥ 0. Moreover, we derive the multiplicity formulas, both in recursive form and in closed form, for each irreducible component appearing in the S 3 -module decomposition of homogeneous subspaces. [ABSTRACT FROM AUTHOR]

It is a well-known result of Auslander and Reiten that contravariant finiteness of the class P ∞ fin (of finitely generated modules of finite projective dimension) over an Artin algebra is a sufficient condition for validity of finitistic dimension conjectures. Motivated by the fact that finitistic dimensions of an algebra can alternatively be computed by Gorenstein projective dimension, we examine in this work the Gorenstein counterpart of Auslander–Reiten condition, namely contravariant finiteness of the class GP ∞ fin (of finitely generated modules of finite Gorenstein projective dimension), and its relation to validity of finitistic dimension conjectures. It is proved that contravariant finiteness of the class GP ∞ fin implies validity of the second finitistic dimension conjecture over left artinian rings. In the more special setting of Artin algebras, however, it is proved that the Auslander–Reiten sufficient condition and its Gorenstein counterpart are virtually equivalent in the sense that contravariant finiteness of the class GP ∞ fin implies contravariant finiteness of the class P ∞ fin over any Artin algebra, and the converse holds for Artin algebras over which the class GP 0 fin (of finitely generated Gorenstein projective modules) is contravariantly finite. [ABSTRACT FROM AUTHOR]

Dominant dimension is introduced into integral representation theory, extending the classical theory of dominant dimension of Artinian algebras to projective Noetherian algebras (that is, algebras which are finitely generated projective as modules over a commutative Noetherian ring). This new homological invariant is based on relative homological algebra introduced by Hochschild in the 1950s. Amongst the properties established here are a relative version of the Morita-Tachikawa correspondence and a relative version of Mueller's characterization of dominant dimension. The behaviour of relative dominant dimension of projective Noetherian algebras under change of ground ring is clarified and we explain how to use this property to determine the relative dominant dimension of projective Noetherian algebras. In particular, we determine the relative dominant dimension of Schur algebras and quantized Schur algebras. [ABSTRACT FROM AUTHOR]

We study the double centraliser algebras and annihilator ideals of certain permutation modules for symmetric groups and their quantum analogues. Of particular interest is the behaviour of these algebras and ideals under base change. By exploiting connections with polynomial representation theory over a field we establish the double centraliser property and dimension invariance of annihilator ideals over fields. By elementary localisation techniques we obtain the corresponding results over arbitrary rings. The modules considered include the permutation modules relevant to partition algebras, as studied in the recent paper by Bowman, Doty and Martin, [3] , and we obtain in particular their main results on the double centraliser property in this context. Much of our development is equally valid for the corresponding modules over Hecke algebras of type A and, for the most part, we give it at this level of generality. [ABSTRACT FROM AUTHOR]

We prove that a sequence h of non-negative integers is the Hilbert function of some Artinian Gorenstein algebra with the strong Lefschetz property (SLP) if and only if it is an Stanley-Iarrobino-sequence. This generalizes the result by T. Harima which characterizes the Hilbert functions of Artinian Gorenstein algebras with the weak Lefschetz property. We also provide classes of Artinian Gorenstein algebras obtained from the ideal of points in \mathbb {P}^n such that some of their higher Hessians have non-vanishing determinants. Consequently, we provide families of such algebras satisfying the SLP. [ABSTRACT FROM AUTHOR]

In this article, we present a trichotomy (a division into three classes) on Noetherian and Artinian C*-algebras and obtain some structural results about Noetherian (and/or Artinian) C*-algebras. We show that every Noetherian, purely infinite and σ-unital C*-algebra A is generated as a C*-ideal by a single projection. We show that if A is a purely infinite, nuclear, separable, Noetherian and Artinian C*-algebra, then A ≅ A ⊗ Z ≅ A ⊗ O ∞ . This is a partial extension of Kirchberg's O ∞ -absorption theorem and Kirchberg's exact embedding theorem. Finally, we show that each Noetherian AF-algebra has a full finite-dimensional C*-subalgebra. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring and M a nonzero R -module. We introduce the class of pseudo-strongly (PS)-hollow submodules of M. Inspired by the theory of modules with secondary representations, we investigate modules which can be written as finite sums of PS-hollow submodules. In particular, we provide existence and uniqueness theorems for the existence of minimal PS-hollow strongly representations of modules over Artinian rings. [ABSTRACT FROM AUTHOR]

In this article, we study the k -Lefschetz properties for non-Artinian algebras, proving that several known results in the Artinian case can be generalized in this setting. Moreover, we describe how to characterize the graded algebras having the k -Lefschetz properties using sectional matrices. We then apply the obtained results to the study of the Jacobian algebra of hyperplane arrangements, with particular attention to the class of free arrangements. [ABSTRACT FROM AUTHOR]

Let k be a field of characteristic zero, n < N be positive integers, P be the polynomial ring k [ x , y , z , w ] , F be the homogeneous polynomial x n + y n + z n + w n , K be the ideal (x N , y N , z N , w N) , and P ‾ be the hypersurface ring P ‾ = P / (F). We describe the minimal multi-homogeneous resolution of P ‾ / K P ‾ by free P ‾ -modules, the socle degrees of P ‾ / K P ‾ , and the minimal multi-homogeneous resolution of the Gorenstein ring P / (K : F) by free P -modules. Our arguments use Stanley's theorem that every Artinian monomial complete intersection over a polynomial ring with coefficients from a field of characteristic zero has the strong Lefschetz property as well as a multi-grading on P for which both ideals K and (F) are homogeneous. The resolution of P ‾ / K P ‾ by free P ‾ -modules is obtained from a Differential Graded Algebra resolution of P / (K : F) by free P -modules, together with one homotopy map. The multi-grading is used to prove that the resulting resolution is minimal. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring and U (R) the multiplicative group of unit elements of R. In 2012, Khashyarmanesh et al. defined the generalized unit and unitary Cayley graph, Γ (R , G , S) , corresponding to a multiplicative subgroup G of U (R) and a nonempty subset S of G with S − 1 = { s − 1   |   s ∈ S } ⊆ S , as the graph with vertex set R and two distinct vertices x and y being adjacent if and only if there exists s ∈ S such that x + s y ∈ G. In this paper, we characterize all Artinian rings R for which Γ (R , U (R) , S) is projective. This leads us to determine all Artinian rings whose unit graphs, unitary Cayley graphs and co-maximal graphs are projective. In addition, we prove that for an Artinian ring R for which Γ (R , U (R) , S) has finite nonorientable genus, R must be a finite ring. Finally, it is proved that for a given positive integer k , the number of finite rings R for which Γ (R , U (R) , S) has nonorientable genus k is finite. [ABSTRACT FROM AUTHOR]

In this paper, we show that if M is a non-zero Artinian R-module and x := x1;...; xn is anM-coregular sequence, then x1;...; xn is a D(Hx n(M))-coregular sequence. Moreover, if R is complete with respect to I-adic topology and d = NdimM, then dimHI d (M) = d and depthHd I (M) = minf2; dg whenever HI d (M) 6= 0:. [ABSTRACT FROM AUTHOR]

We discuss some variants of ascending and descending chain conditions (see [2], [3] and [8] ) analogously. We introduce the idea of nem-Noetherian and nem-Artinian modules and rings. A right R-module M is said to be nem-Noetherian (nem-Artinian) if for every ascending (descending) chain M1 M2 M3: : : (M1 M2 M3 : : :) of non-essential submodules of M, there exists an index n such that Mi+1 embeds in Mi (Mi embeds in Mi+1), for every i n. We characterize these modules such that M is nem-Artinian (nem-Noetherian) if and only if every non-essential submodule of M is mono-Artinian (mono-Noetherian) if and only if every proper closed submodule of M is mono-Artinian (mono-Noetherian). Also, we study several properties of these modules. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with nonzero identity and M be an R-module. In this paper, we introduce the concept of r-Artinian modules which is a new generalization of Artinian modules. An R-module M is called an r-Artinian module if M satisfies the descending chain condition on r-submodules. Also, we call the ring R to be an r-Artinian ring if R is an r-Artinian R-module. We prove that an R-module M is an r-Artinian module if and only if its total quotient module is an Artinian module. In particular, we observe that r-Artinian modules generalize S-Artinian modules, for some particular multiplicatively closed subsets S of R. Also, we extend many properties of Artinian modules to r-Artinian modules. Finally, we use the idealization construction to give non-trivial examples of r-Artinian rings that are not Artinian. [ABSTRACT FROM AUTHOR]

We introduce the notion of a rank function on a triangulated category 𝒞 {\mathcal{C}} which generalizes the Sylvester rank function in the case when 𝒞 = 𝖯𝖾𝗋𝖿 ⁢ (A) {\mathcal{C}=\mathsf{Perf}(A)} is the perfect derived category of a ring A. We show that rank functions are closely related to functors into simple triangulated categories and classify Verdier quotients into simple triangulated categories in terms of particular rank functions called localizing. If 𝒞 = 𝖯𝖾𝗋𝖿 ⁢ (A) {\mathcal{C}=\mathsf{Perf}(A)} as above, localizing rank functions also classify finite homological epimorphisms from A into differential graded skew-fields or, more generally, differential graded Artinian rings. To establish these results, we develop the theory of derived localization of differential graded algebras at thick subcategories of their perfect derived categories. This is a far-reaching generalization of Cohn's matrix localization of rings and has independent interest. [ABSTRACT FROM AUTHOR]

We introduce the concept of Z-symmetric rings. In fact, the classes of all semicommutative rings, nil rings, reduced rings, Artinian rings and eversible rings are Z-symmetric rings. In order to sustain our assertion, we provide a number of examples of Z-symmetric and non Z-symmetric rings. We observe that the class of Z-symmetric rings lies strictly between the classes of eversible rings and the Dedekind finite rings. In particular, we consider the extensions of Z-symmetric rings. Finally, some new results between the Z-symmetric rings and Armendariz rings will be explored and investigated. [ABSTRACT FROM AUTHOR]

In this paper, by taking the class of all C 3 (or D 3) right R -modules for general envelopes and covers, we characterize a semisimple artinian ring (or a right perfect ring) via D 3 -covers (or D 3 -envelopes) and a right V -ring (or a right noetherian V -ring) via C 3 -covers (or C 3 -envelopes). By using isosimple-projective preenvelope, we obtained that if R is a semiperfect right FGF ring (or left coherent ring), then every isosimple right R -module has a projective cover. Moreover, we also characterize semihereditary serial rings (respectively, hereditary artinian serial rings) in terms of epic flat (respectively, projective) envelopes. [ABSTRACT FROM AUTHOR]

Let R be a commutative Artinian ring and let Γ E (R) be the compressed zero-divisor graph associated to R. The question of when the clique number ω (Γ E (R)) < ∞ was raised by J. Coykendall, S. Sather-Wagstaff, L. Sheppardson, and S. Spiroff. They proved that if ℓ (R) ≤ 4 (where ℓ (R) is the largest length of any of its chains of ideals), then ω (Γ E (R)) < ∞. When ℓ (R) = 6 , they gave an example of a local ring R where ω (Γ E (R)) = ∞ is possible by using the trivial extension of an Artinian local ring by its dualizing module. The question of what happens when ℓ (R) = 5 was stated as an open question. We show that if ℓ (R) = 5 then ω (Γ E (R)) < ∞. We first reduce the problem to the case of a local ring (R , m , k). We then enumerate all possible Hilbert functions of R and show that the k-vector space m / m 2 admits a symmetric bilinear form in some cases of the Hilbert function. This allows us to relate the orthogonality in the bilinear space m / m 2 with the structure of zero-divisors in R. For instance, in the case when m 2 is principal and m 3 = 0 , we show that R is Gorenstein if and only if the symmetric bilinear form on m / m 2 is non-degenerate. Moreover, in the case when ℓ (R) = 4 , our techniques also yield a simpler and shorter proof of the finiteness of ω (Γ E (R)) avoiding, for instance, the Cohen structure theorem. [ABSTRACT FROM AUTHOR]

A ring is called nil clean if every element can be written as the sum of an idempotent and a nilpotent. The class of nil clean rings has emerged as an important variant of the much-studied class of clean rings. However, the nil clean definition is fairly restrictive, and the collection of examples is rather constrained. In this paper, we propose an ideal-theoretic version of the nil clean concept, with the goal of creating a more expansive class of rings. Specifically, we define a ring to be ideally nil clean (INC) if every (two-sided) ideal can be written as the sum of an idempotent ideal and a nil ideal. By passing from elements to ideals, we vastly expand the collection of rings under consideration. For example, all artinian rings and all von Neumann regular rings are INC. We show that, in the commutative case, the INC rings are exactly the strongly π-regular rings. We also explore the relationship between the nil clean and INC properties. After establishing a robust collection of examples of INC rings, we explore the behavior of the INC condition under common ring extensions. In addition, we state several open questions and suggest areas for further investigation. [ABSTRACT FROM AUTHOR]

Sets of zero-dimensional ideals in the polynomial ring k x , y that share the same leading term ideal with respect to a given term ordering are known to be affine spaces called Gröbner cells. Conca-Valla and Constantinescu parametrize such Gröbner cells in terms of certain canonical Hilbert-Burch matrices for the lexicographical and degree-lexicographical term orderings, respectively. In this paper, we give a parametrization of (x , y) -primary ideals in Gröbner cells which is compatible with the local structure of such ideals. More precisely, we extend previous results to the local setting by defining a notion of canonical Hilbert-Burch matrices of zero-dimensional ideals in the power series ring k 〚 x , y 〛 with a given leading term ideal with respect to a local term ordering. [ABSTRACT FROM AUTHOR]

In this paper, we study rings with the property that every cyclic module is almost-injective (CAI). It is shown that R is an Artinian serial ring with J (R) 2 = 0 if and only if R is a right CAI-ring with the finitely generated right socle (or I-finite) if and only if every semisimple right R-module is almost injective, R R is almost injective and has finitely generated right socle. Especially, R is a two-sided CAI-ring if and only if every (right and left) R-module is almost injective. From this, we have the decomposition of a CAI-ring via an SV-ring for which Loewy (R) ≤ 2 and an Artinian serial ring whose squared Jacobson radical vanishes. We also characterize a Noetherian right almost V-ring via the ring for which every semisimple right R-module is almost injective. [ABSTRACT FROM AUTHOR]

An FI- or an OI-module M over a corresponding noetherian polynomial algebra P} may be thought of as a sequence of compatible modules Mn over a polynomial ring Pn whose number of variables depends linearly on n. In order to study invariants of the modules Mn in dependence of n, an equivariant Hilbert series is introduced if M is graded. If M is also finitely generated, it is shown that this series is a rational function. Moreover, if this function is written in reduced form rather precise information about the irreducible factors of the denominator is obtained. This is key for applications. It follows that the Krull dimension of the modules Mn grows eventually linearly in n, whereas the multiplicity of Mn grows eventually exponentially in n. Moreover, for any fixed degree j, the vector space dimensions of the degree j components of Mn grow eventually polynomially in n. As a consequence, any graded Betti number of Mn in a fixed homological degree and a fixed internal degree grows eventually polynomially in n. Furthermore, evidence is obtained to support a conjecture that the Castelnuovo-Mumford regularity and the projective dimension of Mn both grow eventually linearly in n. It is also shown that modules M whose width n components Mn are eventually Artinian can be characterized by their equivariant Hilbert series. Using regular languages and finite automata, an algorithm for computing equivariant Hilbert series is presented. [ABSTRACT FROM AUTHOR]

We prove the existence of a natural subclass of subdirectly indecomposable algebras all of whose Hausdorff numberings are negative. We also construct a T1-separable nonnegative subdirectly indecomposable algebra with Artinian congruence lattice. [ABSTRACT FROM AUTHOR]

We study the class of virtually homo-uniserial modules and rings as a nontrivial generalization of homo-uniserial modules and rings. An R-module M is virtually homo-uniserial if, for any finitely generated submodules 0 ≠ K , L ⊆ M , the factor modules K / Rad (K) , and L / Rad (L) are virtually simple and isomorphic (an R-module M is virtually simple if, M ≠ 0 and M ≅ N for every nonzero submodule N of M). Also, an R-module M is called virtually homo-serial if it is a direct sum of virtually homo-uniserial modules. We obtain that every left R-module is virtually homo-serial if and only if R is an Artinian principal ideal ring. Also, it is shown that over a commutative ring R, every finitely generated R-module is virtually homo-serial if and only if R is a finite direct product of almost maximal uniserial rings and principal ideal domains with zero Jacobson radical. Finally, we obtain some structure theorems for commutative (Noetherian) rings whose every proper ideal is virtually (homo-)serial. [ABSTRACT FROM AUTHOR]

The Schröder-Bernstein Theorem for sets is well-known. A dual of the Schröder-Bernstein Theorem is that two sets with surjection maps onto each other are isomorphic. Analogous to this dual, the question of whether two algebraic structures which are epimorphic to each other, are always isomorphic to each other, is of interest. For modules over a given ring, this does not hold true in general. For a ring R , a subclass C of R -modules is said to satisfy the dual of Schröder-Bernstein (or DSB) property if any pair of its members are isomorphic whenever each one is epimorphic image of the other. We investigate the DSB property for the classes of (quasi-)discrete and (quasi-)projective modules among other results in this paper. In particular, we prove that the class of discrete R -modules has the DSB property while the class of quasi-discrete modules does not satisfy the DSB property. On the other hand, over Dedekind domains and generalized uniserial rings, the DSB property holds for the class of quasi-discrete modules. We show that over a right (semi-)perfect ring R , the class of (finitely generated) quasi-projective R -modules satisfies the DSB property, however, over a (von-Neumman) regular ring R these classes do not satisfy the property. We also investigate the DSB property for the class of injective modules. As applications, our investigations provide answers to several open questions posed earlier in this regard. [ABSTRACT FROM AUTHOR]

A moduleM is called ss-lifting if for every submodule A of M, there is a decomposition M = M1 ⊕ M2 such that M1 ≤ A and A∩M2 ⊆ Socs (M), where Socs(M) = Soc(M)∩Rad(M). In this paper, we provide the basic properties of ss-lifting modules. It is shown that: (1) a module M is ss-lifting iff it is amply ss-supplemented and its ss-supplement submodules are direct summand; (2) for a ring R, RR is ss-lifting if and only if it is ss-supplemented iff it is semiperfect and its radical is semisimple; (3) a ring R is a left and right artinian serial ring and Rad (R) ⊆ Soc(RR) iff every left R-module is ss-lifting. We also study on factor modules of ss-lifting modules. [ABSTRACT FROM AUTHOR]

Let R be the power series ring or the polynomial ring over a field k and let I be an ideal of R. Macaulay proved that the Artinian Gorenstein k-algebras R/I are in one-to-one correspondence with the cyclic R-submodules of the divided power series ring \Gamma. The result is effective in the sense that any polynomial of degree s produces an Artinian Gorenstein k-algebra of socle degree s. In a recent paper, the authors extended Macaulay's correspondence characterizing the R-submodules of Γ in one-to-one correspondence with Gorenstein d-dimensional k-algebras. However, these submodules in positive dimension are not finitely generated. Our goal is to give constructive and finite procedures for the construction of Gorenstein k-algebras of dimension one and any codimension. This has been achieved through a deep analysis of the G-admissible submodules of Γ. Applications to the Gorenstein linkage of zero-dimensional schemes and to Gorenstein affine semigroup rings are discussed. [ABSTRACT FROM AUTHOR]

The object of this article is associate to automorphism-invariant modules that are invariant under any automorphism of their injective hulls with cyclic modules and cyclic modules have cyclic automorphism-invariant hulls. The study of the first sequence allows us to characterize rings whose cyclic right modules are automorphism-invariant and to show that if R is a right Köthe ring, then R is an Artinian principal left ideal ring in case every cyclic right R-module is automorphism-invariant. The study of the second sequence leads us to consider a generalization of hypercyclic rings that are each cyclic R-module has a cyclic automorphism-invariant hull. Such rings are called right a-hypercyclic rings. It is shown that every right a-hypercyclic ring with Krull dimension is right Artinian. [ABSTRACT FROM AUTHOR]

Let R be an unital ring which is not necessarily commutative. The intersection graph of ideals of R is a graph with the vertex set which contains proper ideals of R and distinct two vertices I and J are adjacent if and only if I Ç J ¹ 0 is denoted by G. In this paper, we will give some properties of regular graph, triangle-free graph and clique number of G(M) for a module M. We also characterize girth of an Artinian module with connected module. We characterize the chromatic number of G(Zn). We also give an algorithm for the chromatic number of G(Zn). [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity. In this paper, a Cohen-type theorem for w -Artinian modules is given, i.e. a w -cofinitely generated R -module M is w -Artinian if and only if (M / ann M (P)) w is w -cofinitely generated for every prime w -ideal P of R. As a by-product of the proof, we also obtain a detailed representation of elements of a w -module and the w -theoretic version of the Chinese remainder theorem for both modules and rings. [ABSTRACT FROM AUTHOR]

Weakening the notion of R -projectivity, a right R -module M is called max-projective provided that each homomorphism f : M → R / I , where I is any maximal right ideal, factors through the canonical projection π : R → R / I. We study and investigate properties of max-projective modules. Several classes of rings whose injective modules are R -projective (respectively, max-projective) are characterized. For a commutative Noetherian ring R , we prove that injective modules are R -projective if and only if R = A × B , where A is Q F and B is a small ring. If R is right hereditary and right Noetherian then, injective right modules are max-projective if and only if R = S × T , where S is a semisimple Artinian and T is a right small ring. If R is right hereditary then, injective right modules are max-projective if and only if each injective simple right module is projective. Over a right perfect ring max-projective modules are projective. We discuss the existence of non-perfect rings whose max-projective right modules are projective. [ABSTRACT FROM AUTHOR]

In this paper we study modules with the WFI+-extending property. We prove that if M satisfies the WFI+-extending, pseudo duo properties and M=(SocM) has finite uniform dimension then M decompose into a direct sum of a semisimple submodule and a submodule of finite uniform dimension. In particular, if M satisfies the WFI+- extending, pseudo duo properties and ascending chain (respectively, descending chain) condition on essential submodules then M = M1 M2 for some semisimple submodule M1 and Noetherian (respectively, Artinian) submodule M2. Moreover, we show that if M is a WFI-extending module with pseudo duo, C2 and essential socle then the quotient ring of its endomorphism ring with Jacobson radical is a (von Neumann) regular ring. We provide several examples which illustrate our results. [ABSTRACT FROM AUTHOR]

G.W. Mackey's celebrated obstruction theory for projective representations of locally compact groups was remarkably generalized by J. M. G. Fell and R. S. Doran to the wide area of saturated Banach *-algebraic bundles. Analogous obstruction is suggested here for discrete group graded algebras which are not necessarily saturated, i.e. strongly graded in the discrete context. The discrete obstruction is a map assigning a certain second cohomology class to every equivariance class of absolutely graded-simple modules. The set of equivariance classes of such modules is equipped with an appropriate multiplication, namely a graded product, such that the obstruction map is a homomorphism of abelian monoids. Graded products, essentially arising as pull-backs of bundles, admit many nice properties, including a way to twist graded algebras and their graded modules. The obstruction class turns out to determine the fine part that appears in the Bahturin-Zaicev-Sehgal decomposition, i.e. the graded Artin-Wedderburn theorem for graded-simple algebras which are graded Artinian, in case where the base algebras (i.e. the unit fiber algebras) are finite-dimensional over algebraically closed fields. [ABSTRACT FROM AUTHOR]

This document is an erratum for an article titled "Left Co-Köthe Rings and Their Characterizations" published in the journal Communications in Algebra. The erratum corrects several errors in the original article, including deleting certain conditions and statements, and making revisions to theorems and lemmas. The corrected version provides accurate information about left and right R-modules, Kawada rings, and the commutation of maximal ideals in rings. The corrections are supported by references to previous propositions and theorems. The authors of the erratum are Shadi Asgari, Mahmood Behboodi, and Somayeh Khedrizadeh. [Extracted from the article]

In (Stanley, 1978), Stanley constructs an example of an Artinian Gorenstein (AG) ring A with non-unimodal H -vector (1 , 13 , 12 , 13 , 1). Migliore-Zanello show in (Migliore and Zanello, 2017) that for regularity r = 4 , Stanley's example has the smallest possible codimension c for an AG ring with non-unimodal H -vector. The weak Lefschetz property (WLP) has been much studied for AG rings; it is easy to show that an AG ring with non-unimodal H -vector fails to have WLP. In codimension c = 3 it is conjectured that all AG rings have WLP. For c = 4 , Gondim shows in (Gondim, 2017) that WLP always holds for r ≤ 4 and gives a family where WLP fails for any r ≥ 7 , building on Ikeda's example (Ikeda, 1996) of failure for r = 5. In this note we study the minimal free resolution of A and relation to Lefschetz properties (both weak and strong) and Jordan type for c = 4 and r ≤ 6. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring, α an ideal of R and M an R-module with dim M = d. We get equivalent conditions for top local cohomology module Hdα(M) to be Artinian and α-cofinite Artinian separately. In addition, we prove that if (R, m) is a local ring such that ExtiR(R/α, M) is minimax, for each i ≤ d, then ExtiR(N,M) is minimax R-module for each i ≤ 0 and for each finitely generated R-module N with dimN ≤ 2 and SuppR(N) ⊆ V(a). As a consequence we prove that if dimR/α = 2 and SuppR(M) ⊆ V(α), then M is α-cominimax if (and only if) HomR(R/α, M), Ext1R(R/α, M) and Ext2R(R/α,M) are minimax. We also prove that if dim R/α=2 and n ∈ N0 such that ExtiR(R/a, M) is minimax for all i ≤ n+1, then Hiα(M) is α-cominimax for all i < n if (and only if) HomR(R/a,Hiα(M)) is minimax for all i ≤ n. [ABSTRACT FROM AUTHOR]

Let S be a deeply embedded, equicharacteristic, Artinian Gorenstein local ring. We prove that if R is a non-Gorenstein quotient of S of small colength, then every totally reflexive R -module is free. Indeed, the second syzygy of the canonical module of R has a direct summand T which is a test module for freeness over R in the sense that if Tor + R (T , N) = 0 , for some finitely generated R -module N , then N is free. [ABSTRACT FROM AUTHOR]

For any commutative ring A, we introduce a generalization of S-artinian rings using a hereditary torsion theory σ instead of a multiplicative closed subset S ⊆ A. It is proved that if A is a totally σ-artinian ring, then σ must be of finite type, and A is totally σ-noetherian. [ABSTRACT FROM AUTHOR]

We apply minimal weakly generating sets to study the existence of Add (U R) -covers for a uniserial module U R. If U R is a uniserial right module over a ring R , then S : = End (U R) has at most two maximal (right, left, two-sided) ideals: one is the set I of all endomorphisms that are not injective, and the other is the set K of all endomorphisms of U R that are not surjective. We prove that if U R is either finitely generated, or artinian, or I ⊂ K , then the class Add (U R) is covering if and only if it is closed under direct limit. Moreover, we study endomorphism rings of artinian uniserial modules giving several examples. [ABSTRACT FROM AUTHOR]

A Noetherian (Artinian) Lie algebra satisfies the maximal (minimal) condition for ideals. Generalisations include quasi-Noetherian and quasi-Artinian Lie algebras. We study conditions on prime ideals relating these properties. We prove that the radical of any ideal of a quasi-Artinian Lie algebra is the intersection of finitely many prime ideals, and an ideally finite Lie algebra is quasi-Noetherian if and only if it is quasi-Artinian. Both properties are equivalent to soluble-by-finite. We also prove a structure theorem for serially finite Artinian Lie algebras. [ABSTRACT FROM AUTHOR]

The main aim of this article is to study some special elements (zero divisors and units) in ternary rings. Then, the main properties and concepts of noetherian and artinian ternary rings have been studied. In addition, new results on noetherian and artinian ternary rings have been investigated. [ABSTRACT FROM AUTHOR]

We introduce Rad-discrete and quasi-Rad-discrete modules as a proper generalization of (quasi) discrete modules, and provide various properties of these modules. We prove that a direct summand of a (quasi) Rad-discrete module is (quasi) Rad-discrete. We show that every projective R-module is (quasi) Rad-discrete if and only if R is left perfect. We also prove that, over a commutative Noetherian ring R, every quasi-Rad-discrete R-module is the direct sum of local R-modules if and only if R is Artinian. Finally, we investigate self-projective Rad-discrete modules and π -projective quasi-Rad-discrete modules over Dedekind domains. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and I ∗ (R) , the set of all nontrivial proper ideals of R. The intersection graph of ideals of R , denoted by ℐ (R) , is a simple undirected graph with vertex set as the set I ∗ (R) , and, for any two distinct vertices I and J are adjacent if and only if I ∩ J ≠ (0). In this paper, we study some connections between commutative ring theory and graph theory by investigating topological properties of intersection graph of ideals. In particular, it is shown that for any nonlocal Artinian ring R , ℐ (R) is a projective graph if and only if R ≅ R 1 × F 1 , where R 1 is a local principal ideal ring with maximal ideal 𝔪 1 of nilpotency three and F 1 is a field. Furthermore, it is shown that for an Artinian ring R , γ ¯ (ℐ (R)) = 2 if and only if R ≅ R 1 × R 2 , where each R i (1 ≤ i ≤ 2) is a local principal ideal ring with maximal ideal 𝔪 i ≠ (0) such that 𝔪 i 2 = (0). [ABSTRACT FROM AUTHOR]

Recently, in a series of papers "simple" versions of direct-injective and direct-projective modules have been investigated. These modules are termed as "simple-direct-injective" and "simple-direct-projective," respectively. In this paper, we give a complete characterization of the aforementioned modules over the ring of integers and over semilocal rings. The ring is semilocal if and only if every right module with zero Jacobson radical is simple-direct-projective. The rings whose simple-direct-injective right modules are simple-direct-projective are fully characterized. These are exactly the left perfect right H-rings. The rings whose simple-direct-projective right modules are simple-direct-injective are right max-rings. For a commutative Noetherian ring, we prove that simple-direct-projective modules are simple-direct-injective if and only if simple-direct-injective modules are simple-direct-projective if and only if the ring is Artinian. Various closure properties and some classes of modules that are simple-direct-injective (resp. projective) are given. [ABSTRACT FROM AUTHOR]

We first obtain that NI rings satisfy a property that if ab is central for elements a, b, then (a b) n = (b a) n for some n ≥ 1 , by applying a property of reduced rings. We prove next the following: Let R be a ring and I be the ideal of R generated by the subset { a b − b a | a , b ∈ R such that a b is central in R } . (i) Suppose that ab is central for a , b ∈ R and ab – ba is a nonzero nilpotent. Then, A (a b − b a) A is a nonzero nilpotent ideal of the subring A of R, where 1 is the identity of R, B = Z · 1 = { n 1 | n ∈ Z } , and A is the algebra B 〈 a , b 〉 generated by a, b over B. (ii) If R is NI, then I is nil and R/I is an Abelian NI ring. (iii) Let R be reversible and ab be central for a , b ∈ R . Then, there exists l ≥ 1 such that, for every n ≥ l , (a b) n = (b a) n and (a b) n = b h (a b) n − h a h for all 1 ≤ h ≤ n ; especially a n b n = (a b) n = b n a n . We call a ring pseudo-NI if it satisfies the first property of NI rings to be mentioned and examine the structures of NI and pseudo-NI rings in several ring theoretic situations, showing that semisimple Artinian rings are pseudo-NI. [ABSTRACT FROM AUTHOR]

We provide a microlocal necessary condition for distinction of admissible representations of real reductive groups in the context of spherical pairs. Let ${\mathbf {G}}$ be a complex algebraic reductive group and ${\mathbf {H}}\subset {\mathbf {G}}$ be a spherical algebraic subgroup. Let ${\mathfrak {g}},{\mathfrak {h}}$ denote the Lie algebras of ${\mathbf {G}}$ and ${\mathbf {H}}$ , and let ${\mathfrak {h}}^{\bot }$ denote the orthogonal complement to ${\mathfrak {h}}$ in ${\mathfrak {g}}^*$. A ${\mathfrak {g}}$ -module is called ${\mathfrak {h}}$ -distinguished if it admits a nonzero ${\mathfrak {h}}$ -invariant functional. We show that the maximal ${\mathbf {G}}$ -orbit in the annihilator variety of any irreducible ${\mathfrak {h}}$ -distinguished ${\mathfrak {g}}$ -module intersects ${\mathfrak {h}}^{\bot }$. This generalises a result of Vogan [Vog91]. We apply this to Casselman–Wallach representations of real reductive groups to obtain information on branching problems, translation functors and Jacquet modules. Further, we prove in many cases that – as suggested by [Pra19, Question 1] – when H is a symmetric subgroup of a real reductive group G, the existence of a tempered H-distinguished representation of G implies the existence of a generic H-distinguished representation of G. Many of the models studied in the theory of automorphic forms involve an additive character on the unipotent radical of the subgroup $\bf H$ , and we have devised a twisted version of our theorem that yields necessary conditions for the existence of those mixed models. Our method of proof here is inspired by the theory of modules over W-algebras. As an application of our theorem we derive necessary conditions for the existence of Rankin–Selberg, Bessel, Klyachko and Shalika models. Our results are compatible with the recent Gan–Gross–Prasad conjectures for nongeneric representations [GGP20]. Finally, we provide more general results that ease the sphericity assumption on the subgroups, and apply them to local theta correspondence in type II and to degenerate Whittaker models. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity. The co-annihilating-ideal graph of R, denoted by AR, is a graph whose vertex set is the set of all non-zero proper ideals of R and two distinct vertices I and J are adjacent whenever Ann(I) ∩ Ann(J) = (0). In this paper, we characterize all Artinian rings for which both of the graphs AR and AR (the complement of AR), are chordal. Moreover, all Artinian rings whose AR (and thus AR) is perfect are characterized. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring and M be a finitely generated R-module. Let I(M) be the first nonzero Fitting ideal of M. The main result of this paper is to characterize modules whose first nonzero Fitting ideals are prime ideals, in some cases. As a consequence, it is shown that if M is an Artinian R-module and I (M) = q is a prime ideal of R which contains a nonzero divisor, then M ≅ R / q ⊕ P , for some submodule P of M. [ABSTRACT FROM AUTHOR]