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The notion of local rings with quasi-decomposable maximal ideal was formally introduced by Nasseh and Takahashi. In separate works, the authors of the present paper showed that such rings have rigid homological properties; for instance, they are both Ext- and Tor-friendly. One point of this paper is to further explore the homological properties of these rings and also introduce new classes of such rings from a combinatorial point of view. Another point is to investigate how far some of these homological properties can be pushed along certain diagrams of local ring homomorphisms. [ABSTRACT FROM AUTHOR]

For a graph G = (V , E) the edge ring k [ G ] is k [ x 1 , ... , x n ] / I (G) , where n = | V | and I(G) is generated by { x i x j ; { i , j } ∈ E } . The conjecture we treat is the following. Conjecture 1. If k [ G ] has a 2-linear resolution, then the projective dimension of K [ G ] , pd (k [ G ]) , equals the maximal degree of a vertex in G. As far as we know, this conjecture is first mentioned in a paper by Gitler and Valencia [7, Conjecture 4.13], and there it is called the Eliahou-Villarreal conjecture. The conjecture is treated in a recent paper by Ahmed, Mafi, and Namiq [1]. That there are counterexamples was noted already by Moradi and Kiani [9]. By interpreting k [ G ] as a Stanley-Reisner ring, we are able to characterize those graphs for which the conjecture holds. [ABSTRACT FROM AUTHOR]

For a simple graph G with edge ideal I(G), we study the N -graded Betti numbers in the linear strand of the minimal free resolution of I (Γ (Z n)) , where Γ (Z n) is the zero divisor graph of the ring Z n . We present sharp bounds for the Betti numbers of Γ (Z n) and characterize the graphs attaining these bounds, thereby establishing the correct equality case for one of the results of the earlier published paper (Theorem 4.5, S. Pirzada and S. Ahmad, On the linear strand of edge ideals of some zero divisor graphs, Commun. Algebra 51(2) (2023) 620–632). Also, we present homological invariants of the edge rings of Γ (Z n) for n = p 2 q and pqr, with primes p < q < r. [ABSTRACT FROM AUTHOR]

For a simple graph G with edge ideal I(G), we study the N -graded Betti numbers in the linear strand of the minimal free resolution of I (Γ (Z n)) , where Γ (Z n) is the zero divisor graph of the ring Z n . We present sharp bounds for the Betti numbers of Γ (Z n) and characterize the graphs attaining these bounds, thereby establishing the correct equality case for one of the results of the earlier published paper (Theorem 4.5, S. Pirzada and S. Ahmad, On the linear strand of edge ideals of some zero divisor graphs, Commun. Algebra 51(2) (2023) 620–632). Also, we present homological invariants of the edge rings of Γ (Z n) for n = p 2 q and pqr, with primes p < q < r. [ABSTRACT FROM AUTHOR]

AbstractThe aim of this paper is to construct a one dimensional universal central extension of the solenoidal Lie algebra W(n)μ introduced by Y. Billig and V. Futorny. The obtained Lie algebra is higher rank Virasoro algebra denoted by Vir(n)μ. Moreover, we study Vir(n)μ-Harich-Chandra modules. We obtain two kinds of Harich-Chandra modules, the intermediate series modules and generalized weight modules. Furthermore, we use the triangular decomposition of Vir(n)μ given by lexicographic order on Zn to construct new weight modules having infinite dimensional weight spaces. [ABSTRACT FROM AUTHOR]

In this paper, Whittaker modules are studied for a subalgebra q ϵ of the N =2 superconformal algebra. The Whittaker modules are classified by central characters. Additionally, criteria for the irreducibility of the Whittaker modules are given. [ABSTRACT FROM AUTHOR]

The weight modules of the Lie algebra s l (2 , C) are well known. In the first part of this paper we deal with a realization of weight modules of s l (2 , C) in the space t ν C [ t , t − 1 ] , ν ∈ R , where C [ t , t − 1 ] is the algebra of Laurent polynomials. In the second part, we consider the Hom-Lie algebra s l q (2 , C) of Jackson where q ≠ 0 , 1 . The q-analogue of the above realization in the space t ν C (q) [ t , t − 1 ] , ν ∈ R is considered. We obtain two kinds of q-modules. The regular q-modules which have limits the modules obtained in the classical realization when q goes to 1. The other q-modules have no limits when q goes to 1 and they are called singular modules. [ABSTRACT FROM AUTHOR]

We consider two classes of modules over coherent rings: the Gorenstein FP-injective modules, and the weakly Ding injective modules. The Gorenstein FP-injective modules are the cycles of the exact complexes of FP-injective modules. The weakly Ding injective modules are the cycles of the exact complexes of FP-injective modules that stay exact when applying a functor Hom (A , −) for any FP-injective module A. We prove that the class of Gorenstein FP-injective modules is both covering and preenveloping over any (left) coherent ring with the property that every injective module has finite flat dimension. We also prove that, over the same type of rings, the class of weakly Ding injectives, w D I , is preenveloping in R − Mod . If, moreover, w D I is closed under extensions, then (⊥ w D I , w D I) is a hereditary cotorsion pair. In particular, this is the case when R is a Ding Chen ring, and w D I is closed under extensions. We show that if R is a Ding Chen ring such that every FP-injective module has finite projective dimension, then the two classes of modules coincide. Consequently, the class of weakly Ding injective modules is also covering in this case. In the last part of the paper we prove some analogue results for Gorenstein FP-injective complexes and for weakly Ding injective complexes. [ABSTRACT FROM AUTHOR]

As is well known, a class of 2 n 2 -dimensional Kac-Paljutkin Hopf algebras H 2 n 2 was introduced by Pansera. It is the generalization of the 8-dimensional Kac-Paljutkin Hopf algebra H8. In this paper, the classification of Yetter-Drinfeld modules over H 2 n 2 is given by Radford's method of constructing Yetter-Drinfeld modules for a Hopf algebra. Furthermore, the tensor product of Yetter-Drinfeld modules over H 2 n 2 is established, and the Grothendieck ring r ( H 2 n 2 H 2 n 2 Y D) is described explicitly by generators and relations. Communicated by Julia Plavnik [ABSTRACT FROM AUTHOR]

Edge ideals of finite simple graphs G on n vertices are the ideals I(G) of the polynomial ring S in n variables generated by the quadratic monomials associated with the edges of G. In this paper, we consider the possible pairs of dimensions and depths of S / I (G) for connected graphs with a fixed number of vertices. We discuss such pairs in the case when the dimension is relatively large. As a corollary, we completely determine the pairs for connected graphs with a small number of vertices. We also study the possible pairs for connected chordal graphs. [ABSTRACT FROM AUTHOR]

In this paper, for any prime p, we propose the notion of a p-transitive association scheme. This notion aims to generalize the fact that the regular module of a group algebra of a finite group has a unique principal submodule to the case of the regular modules of modular adjacency algebras of association schemes. We characterize the p-transitive quasi-thin association schemes and the p-transitive association schemes with thin thin residue by their structure theory properties. We also get some results with independent interests. [ABSTRACT FROM AUTHOR]

We study characterizations of sincere modules, sincere silting modules and tilting modules in terms of various vanishing conditions. Let R be a perfect ring and T be an R-module. It is proved that T is sincere silting if and only if T is presilting satisfing the vanishing condition KerExt R 0 ≤ i ≤ 1 (T , −) = 0 , and that T is tilting if and only if KerExt R 0 ⩽ i ⩽ 1 (T , −) = 0 and Gen T ⊆ KerExt R 1 ⩽ i ⩽ 2 (T , −) . As an application, we prove that a sincere silting R-module T of finite projective dimension is tilting if and only if Ext R i (T , T (J)) = 0 for all sets J and all integer i ≥ 1 . This not only extends a main result of Zhang's paper [Self-orthogonal τ-tilting modules and tilting modules, J Pure Appl Algebra, 2022, 226: 106860] from finitely generated modules over Artin algebras to infinitely generated modules over more general rings, but also gives it a different proof without using Auslander-Reiten translations. [ABSTRACT FROM AUTHOR]

The aim of this paper is to describe the irreducible modules for the Jordan algebra of 2 × 2 matrices over an algebraically closed field of characteristic different from 2, 3 and 5 in the class of the commutative power-associative algebras. All irreducible non-unital modules, and irreducible unital modules up to dimension three are classified, namely we find seven non-parameterized and five families of parameterized modules of dimension three. For every k ≥ 2 , an irreducible module of dimension 3k is also constructed. [ABSTRACT FROM AUTHOR]

This paper introduces the notion of dual of automorphism extendable modules. A module M is called automorphism-extendable if for every submodule N of M, every automorphism of N can be extended to an endomorphism of M. We call a module M a dual automorphism-extendable module if whenever K is a submodule of M, then every automorphism ν:M∕K → M∕K lifts to an endomorphism 휃 of M. In this paper we give various examples of dual automorphism-extendable modules and study their properties. In particular, we prove that every dual automorphism-extendable module is a D3-module. It is shown that over a right artinian ring R, an R-module with hollow modules Mi is dual automorphism-extendable if and only if M is quasi-projective. [ABSTRACT FROM AUTHOR]

In this paper we give a complete classification of pointed fusion categories over ℂ of global dimension 8. We first classify the equivalence classes of pointed fusion categories of dimension 8, and then we proceed to determine which of these equivalence classes have equivalent categories of modules following the procedure presented in [9, 11]. The results of this paper permit to recover the classification of twisted quantum doubles of groups of order 8 up to gauge equivalence of braided quasi-Hopf algebras that was previously done in [6] and [5]. [ABSTRACT FROM AUTHOR]

In this paper, we give a method to compute the Gelfand-Kirillov dimensions of some polynomial-type weight modules. These modules are infinite-dimensional irreducible 픬(n,ℂ)-modules and 픭(2n,ℂ)-modules that appeared in the ℤ-graded oscillator generalizations of the classical theorem on harmonic polynomials established by Luo and Xu. We also found that some of these modules have the secondly minimal GK-dimension, and some of them have the larger GK-dimension than the maximal GK-dimension apearing in unitary highest-weight modules. [ABSTRACT FROM AUTHOR]

We state a mild generalization of the classical Schönemann and Eisenstein- Dumas irreducibility criterion inℤ[x] and provide an elementary proof. In the end of the paper, we also provide a concrete example of a polynomial which is irreducible by the main result of the paper but whose irreducibility does not follow from existing criteria. [ABSTRACT FROM PUBLISHER]

Let (R, 𝔪) be a commutative Noetherian complete local ring,Ma nonzero finitely generatedR-module of dimensionn, andIbe an ideal ofR. In this paper we calculate the annihilator of the top local cohomology module. Also, if (R, 𝔪) is a Noetherian local Cohen–Macaulay ring of dimensiondandIis a nonzero proper ideal ofR, then we calculate the annihilator of the first nonzero local cohomology module. Finally, we show that ifRis an arbitrary Noetherian ring,Ian ideal ofR, andMis a nonzero finitely generatedR-module with cd(I,M) = t ≥ 0, then there exists a submoduleNofMsuch that. This is a generalization of the main result of Bahmanpour, A'zami, and Ghasemi [1] for all ideals of an arbitrary Noetherian ringR. [ABSTRACT FROM AUTHOR]

The n-th Schrödinger algebra s c h n : = s l 2 ⋉ h n is the semi-direct product of the Lie algebra s l 2 with the n-th Heisenberg Lie algebra h n . In this paper, we classify the s c h n -modules (n ≥ 2) that are free of rank 1 when restricted to the Cartan subalgebra (modulo center). Furthermore, the simplicities of these modules are determined. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the classifying functors W ¯ and Diag ° N are simplicially homotopy equivalent. Using this equivalency, we define classifying simplicial modules and give an alternative definition of (co)homology of crossed modules of algebras. [ABSTRACT FROM AUTHOR]

Mutation of τ-tilting modules is a basic operation to construct a new support τ-tilting module from a given one by replacing a direct summand. The aim of this paper is to give a positive answer to the question posed in [2, Question 2.31] about mutation of τ-tilting modules. [ABSTRACT FROM AUTHOR]

Two direct-summands A and B of a right R-module M will be called strongly-perspective (s-perspective) if for any X ⊆ M , M = A ⊕ X iff M = B ⊕ X. The module M will be called s-perspective if every two isomorphic direct-summands of M are s-perspective, and a ring R will be called s-perspective if R as a right R-module is s-perspective. In this paper, we study s-perspective rings and modules, and using ideas of Khurana and Nielsen, it follows that if M is s-perspective, then M has the finite exchange property iff M is clean, iff M has the full exchange property. In particular, if M is either summand-square-free or dual-summand-square-free with the finite exchange property, then M is clean, s-perspective and has the full exchange property. [ABSTRACT FROM AUTHOR]

In the recent publication, generalized Nakayama–Azumaya's lemma, i.e. Azumaya's lemma (or Nakayama's lemma) extended to a module which is isomorphic to a direct summand of a direct sum of finitely generated modules, is conjectured to hold. A proof of this conjecture is the main objective of the present paper. It is achieved by proving the nonexistence of non-zero weak Nakayama–Azumaya special modules, and showing that the canonical map f : ⊕ i ∈ N M i → M = ∪ i ∈ N M i is a non-split epimorphism if Mi is finitely generated and M i ⊂ M i + 1 J (R) for each i ∈ N. As an application of generalized Nakayama–Azumaya's lemma, we show the existence of maximal submodules for some classes of modules. This implies, in particular, Bass' result concerning the class of projective modules. [ABSTRACT FROM AUTHOR]

A submodule N of a right R-module M is said to lie over a direct summand of M if, there is a decomposition with and. In this paper, a right R-module M is called a Dual-Utumi-Module (-module) if for any two proper submodules A and B of M with and A + B = M, there exist two summands K and L of M such that A lies over K, B lies over L and Dual-U-modules are strict generalizations of quasi-discrete, pseudo-discrete, and dual-square-free modules. In this paper several characterizations of -modules are provided which in turn are used to show that the class of -modules inherits some of the important features of the aforementioned classes of modules. For example, if is a -module with then M is quasi-projective and discrete. As an immediate application, a ring R is right perfect iff every quasi-projective right R-module is a -module. It is also shown that if M is a -module whose local summands are summands, then where Q and P are factor-orthogonal, Q is P-projective and dual–square-free, is a direct sum of pairwise non-isomorphic indecomposable modules, is quasi-projective and discrete with , and C is isomorphic to a direct summand of. In particular, every quasi-discrete module has such a decomposition. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce the notion of non-abelian tensor product of a given precrossed module by one of its ideals. We use this concept to study the second homology and the nilpotent multiplier of a precrossed module. We also describe the Ganea term for the c-nilpotent multiplier and the universal central extension of precrossed modules in term of the non-abelian tensor product. [ABSTRACT FROM AUTHOR]

Let (R, 𝔪) be a complete local ring, 𝔞 an ideal ofR, andMa finitely generatedR-module. The aim of this paper is to show that for any non-negative integern,, whereis thenth finiteness dimension ofMrelative to 𝔞. As a consequence, it follows that the setis finite. This generalizes the main result of Quy [10] and Brodmann and Lashgari [4]. [ABSTRACT FROM AUTHOR]

A nonzero module M over a commutative ring R is said to have a complete comaximal decomposition if it can be written in the form , where the annihilators of the Ni’s are pairwise comaximal pseudo-irreducible ideals of R. In this paper, we show that a complete comaximal decomposition for an R-module is unique if it exists and a ring R is J-Noetherian if and only if every nonzero R-module has a complete comaximal decomposition. Also, we give a topological characterization of (finitely generated) comultiplication R-modules whose submodules have a complete comaximal decomposition and then we show that a commutative ring R is von Neumann regular if and only if every representable R-module can be written as a finite direct sum of homogeneous semi-simple R-modules. [ABSTRACT FROM AUTHOR]

Let T be a Hochschild extension algebra of a finite dimensional algebra A over a field K by the standard duality A-bimodule HomK(A, K). In this paper, we determine the ordinary quiver of T if A is a self-injective Nakayama algebra by means of the ℕ-graded second Hochschild homology group HH2(A) in the sense of Sköldberg. [ABSTRACT FROM AUTHOR]

With the present note the authors intend to correct a mistake in the paper Semiring and semimodule issues in MV-algebras, Communications in Algebra 41:3 (2013), 1017-1048, by the same authors. [ABSTRACT FROM AUTHOR]

In this paper, we establish an explicit classification of length two extensions of tensor modules for the Witt algebra using the cohomology of the Witt algebra with coefficients in the module of the space of homomorphisms between the two modules of interest. To do this we extended our module to a module that has a compatible action of the commutative algebra of Laurent polynomials in one variable. In this setting, we are be able to directly compute all possible 1-cocycles. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce the concept of "factor-square-full" modules defined as follows: For any proper submodule X of M, there exist a proper submodule Y of M with X ⊆ Y and an epimorphism f : M → (M / Y) 2 , and give some properties of these modules. Using those results, we show that any quasi-discrete module is a direct sum of a factor-square-full module and a dual-square-free module which are dual orthogonal, and consider the problem "When is a quasi-discrete module quasi-projective?". Moreover, we study rings whose factor-square-full modules are closed under essential extensions or essential submodules. [ABSTRACT FROM AUTHOR]

In a paper of the same title Sin [3] has determined the space of extensions between any two irreducible 2-modular representations of the groupsand. The present paper develops a different way to describe such extensions. [ABSTRACT FROM AUTHOR]

The rate of a graded moduleMover a standard graded algebraR,rateR(M), is a measure of the growth of the shifts in the minimal graded free resolution ofM. In this paper, we study the rate of Veronese modules ofM. More precisely, it is shown thatfor allc≥1. This extends a result of Aramova et al. [2] As a consequence of this, ifMis generated in degree zero, then, for allc≥max{rateR(M),Rate(R)}. Also, it is shown that, for all integerss,c≥1. In particular case, we give a simple proof to a result of Backelin. [ABSTRACT FROM PUBLISHER]

In this paper we provide conditions under which automorphism-coinvariant modules over a right perfect ring are quasi-projective. [ABSTRACT FROM PUBLISHER]

A ring R is called left AIP if R modulo the left annihilator of any ideal is flat. In this paper, we characterize a module MR for which the endomorphism ring E n d R (M) is left AIP. We say a module MR is endo-AIP (resp. endo-APP) if M has the property that "the left annihilator in E n d R (M) of every fully invariant submodule of M (resp. E n d R (M) m , for every m ∈ M ) is pure as a left ideal in E n d R (M) ". The notion of endo-AIP (resp. endo-APP) modules generalizes the notion of Rickart and p.q.-Baer modules to a much larger class of modules. It is shown that every direct summand of an endo-AIP (resp.endo-APP) module inherits the property and that every projective module over a left AIP (resp. APP)-ring is an endo-AIP (resp. endo-APP) module. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the non-abelian tensor square of precrossed modules in Lie algebras and investigate some of its properties. In particular, for an arbitrary Lie algebra L, we study the relation of the second homology of a precrossed L-module and the non-abelian exterior square. Also, we show how this non-abelian tensor product is related to the universal central extensions (with respect to the subcategory of crossed modules) of a precrossed module. [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]

Local Weyl modules were originally defined for affine Lie algebras by Chari and Pressley in [5]. In this paper we extend the notion of local Weyl modules for a Lie algebra 𝔤 ⊗A, where 𝔤 is any Kac–Moody algebra and A is any finitely generated commutative associative algebra with unit over ℂ, and prove a tensor product decomposition theorem which generalizes result in [2, 5]. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a subcategory of totally reflexive modules that have a saturated filtration by other totally reflexive modules. We will prove these are precisely the totally reflexive modules with an upper triangular presentation matrix and that such a module has a complete resolution in which every differential can be simultaneously represented by upper triangular matrices. [ABSTRACT FROM AUTHOR]

This article is a sequel to the recent three papers on "virtually semisimple modules and rings," by Behboodi et al., which two of them appeared in the Algebras and Representation Theory and Communications in Algebra in 2018. An R-module M is called virtually semisimple if each submodule of M is isomorphic to a direct summand. A ring R is called left (resp., right) virtually semisimple if (resp., RR) is virtually semisimple. In this article, we study rings and modules in which every prime submodule is isomorphic to a direct summand, and called them prime virtually (or -virtually) semisimple modules. A ring R is called left (resp., right) -virtually semisimple if (resp., RR) is -virtually semisimple. The results of the article are inspired by a characterization of left -virtually semisimple rings. We prove that these rings are precisely the left virtually semisimple rings, and in this case , where each Di is a domain and each is a principal left ideal ring. We also answer to the following questions: (i) Describe rings R where each (finitely generated or cyclic) left R-module is -virtually semisimple?, and (ii) Describe rings R where each left R-module is a direct sum of indecomposable -virtually semisimple modules? Finally, we study -virtually semisimple modules over commutative rings. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in some natural consequences arising from the canonical duality relation between the concepts of groupoid grading and groupoid action. Any algebra A over a field K graded by a finite groupoid has a canonical structure of a module algebra over the dual algebra of the groupoid algebra. The results here presented concern to the multiplicative structure of the smash product algebra and its representations. An application is the affirmative answer to a Bergmann-type question for groupoid gradings. [ABSTRACT FROM AUTHOR]

Let denote an algebraically closed field of characteristic zero and let be some scalars in. By the Racah algebra associated with , we mean the most general quadratic algebra with two algebraically independent generators , which possesses presentations with ladder relations. In this paper, we classify the finite-dimensional irreducible -modules up to isomorphism by using the theory of the Leonard pairs. For a given irreducible -module V with dimension d + 1, we give its corresponding isomorphism classes of Leonard pairs on V that have Racah type. [ABSTRACT FROM AUTHOR]

This paper deals with the variety of commutative algebras satisfying the identity β{(yx2)x − ((yx)x)x} + γ{yx3 − ((yx)x)x} = 0, where β, γ are scalars. These algebras appeared as one of the four families of degree four identities in Carini, Hentzel, and Piacentini-Cattaneo [6]. We give a characterization of representations and irreducible modules on these algebras. Our results require that the characteristic of the ground field is different from 2, 3. [ABSTRACT FROM PUBLISHER]

In this paper, we consider the finite connected quiverQhaving two subquiversQ(1)andQ(2)with. Suppose thatQ(i)is not a subquiver ofQ(j)where {i,j} = {1, 2}. For a monomial algebra Λ =kQ/Iobtained by the quiverQ, when the associated sequence of paths given byMinsharp<(I) satisfies a certain separability condition, we propose the method so that we easily construct a minimal projective resolution of Λ as a right Λe-module and calculate the Hochschild cohomology group of Λ. [ABSTRACT FROM AUTHOR]

In [2] Camillo and Zelmanowitz stated that rings all whose modules are dimension modules are semisimple Artinian. It seem however that the proof in [2] contains a gap and applies to rings with finite Goldie dimension only. In this paper we show that the result indeed holds for all rings with a basis as well as for all commutative rings with Goldie dimension attained. [ABSTRACT FROM AUTHOR]

In the paper we first construct a new cotorsion pair, in the category of chain complexes, from two given cotorsion pairs in the category of modules, and then we consider completeness of such pairs under certain conditions. [ABSTRACT FROM AUTHOR]

Let 𝕂 be a field, and letR= 𝕂[x1,…,xn] be the polynomial ring over 𝕂 innindeterminatesx1,…,xn. LetGbe a graph with vertex-set {x1,…,xn}, and letJbe the cover ideal ofGinR. For a given positive integerk, we denote thekth symbolic power and thekth bracket power ofJbyJ(k)andJ[k], respectively. In this paper, we give necessary and sufficient conditions forR/Jk,R/J(k), andR/J[k]to be Cohen–Macaulay. We also study the limit behavior of the depths of these rings. [ABSTRACT FROM AUTHOR]

In this paper, we give a new simple algebraic proof of the involutivity of the characteristic variety after Gabber, which is one of the most important results ofD-module theory. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of capability for crossed modules of Lie algebras, which is a generalization of capability in Lie algebras and groups. By using a special central ideal of a crossed module, we give a sufficient condition for the capability of a crossed module of Lie algebras. Also, we will extend the five-term exact sequence on homology of crossed modules of Lie algebras one term further and study the connection between the capability of crossed modules and this sequence. Finally, we study the relation between the capability and the center of a cover of a crossed module. [ABSTRACT FROM AUTHOR]

LetCbe a semidualizing module for a commutative ringR. In this paper, we study the resulting modules of finiteGC-projective dimension in Bass class, showing that they admitGC-projective precover. Over local ring, we prove that dimR(M) ≤ 𝒢ℐC − idR(M) for any nonzero finitely generatedR-moduleM, which generalizes a result due to Bass. [ABSTRACT FROM AUTHOR]