On δ-semiperfect modules

A submodule N of a module M is δ-small in M if N+X≠M for any proper submodule X of M with M∕X singular. A projective δ-cover of a module M is a projective module P with an epimorphism to M whose kernel is δ-small in P. A module M is called δ-semiperfect if every factor module of M has a projective δ-cover. In this paper, we prove various properties, including a structure theorem and several characterizations, for δ-semiperfect modules. Our proofs can be adapted to generalize several results of Mares [8 Mares, E. A. (1963). Semi-perfect modules. Math. Z. 82:347–360.[Crossref] , [Google Scholar]] and Nicholson [11 Nicholson, W. K. (1975). On semiperfect modules. Canad. Math. Bull. 18(1):77–80.[Crossref], [Web of Science ®] , [Google Scholar]] from projective semiperfect modules to arbitrary semiperfect modules.     

Almost-Perfect Rings and Modules

Following [1 Amini , A. , Ershad , M. , Sharif , H. ( 2008 ). Rings over which flat covers of finitely generated modules are projective . Comm. Algebra 36 : 2862 – 2871 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]], a ring R is called right almost-perfect if every flat right R-module is projective relative to R. In this article, we continue the study of these rings and will find some new characterizations of them in terms of decompositions of flat modules. Also we show that a ring R is right almost-perfect if and only if every right ideal of R is a cotorsion module. Furthermore, we prove that over a right almost-perfect ring, every flat module with superfluous radical is projective. Moreover, we define almost-perfect modules and investigate some properties of them.     

On Strongly Gorenstein FP-Injective Modules

This article continues to investigate a particular case of Gorenstein FP-injective modules, called strongly Gorenstein FP-injective modules. Some examples are given to show that strongly Gorenstein FP-injective modules lie strictly between FP-injective modules and Gorenstein FP-injective modules. Various results are developed, many extending known results in [1 Bennis , D. , Mahdou , N. ( 2007 ). Strongly Gorenstein projective, injective, and flat modules . J. Pure Appl. Algebra 210 : 437 – 445 .[Crossref], [Web of Science ®] , [Google Scholar]]. We also characterize FC rings in terms of strongly Gorenstein FP-injective, projective, and flat modules.     

Two Open Questions On Goldie Extending Modules

Five open questions on Goldie extending modules were posed by Akalan et al. [1 Akalan , E. , Birkenmeier , G. F. , Tercan , A. ( 2009 ). Goldie extending modules . Comm. Algebra 37 : 663 – 683 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. The first one and the second one are considered in this article. It is shown that a 𝒢-extending module M with (C ₃) is 𝒢⁺-extending. Moreover, let M = M ₁ ⊕ M ₂ be a direct sum of 𝒢-extending modules, where M satisfies (C ₃) and M ₁ is M ₂-ojective (or M ₂ is M ₁-ojective), then M is 𝒢-extending. Other sufficient conditions for a direct sum of 𝒢-extending modules to be 𝒢-extending are obtained.     

Representations of Generalized Almost-Jordan Algebras

This paper deals with the variety of commutative algebras satisfying the identity β{(yx ²)x ? ((yx)x)x} + γ{yx ³ ? ((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 Carini , L. , Hentzel , I. R. , Piacentini-Cattaneo , J. M. ( 1988 ). Degree four identities not implies by commutativity . Comm. in Algebra 16 ( 2 ): 339 – 357 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. 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.     

Feebly Baer Rings and Modules

A right module M over a ring R is called feebly Baer if, whenever xa = 0 with x ∈ M and a ∈ R, there exists e² = e ∈ R such that xe = 0 and ea = a. The ring R is called feebly Baer if R_R is a feebly Baer module. These notions are motivated by the commutative analog discussed in a recent paper by Knox, Levy, McGovern, and Shapiro [6 Knox , M. L. , Levy , R. , McGovern , W. Wm. , Shapiro , J. ( 2009 ) Generalizations of complemented rings with applications to rings of functions. . J. Alg. Appl. 8 ( 1 ): 17 – 40 .[Crossref] , [Google Scholar]]. Basic properties of feebly Baer rings and modules are proved, and their connections with von Neumann regular rings are addressed.     

Modules Which are Isomorphic to Their Factor Modules

Let R be commutative ring with identity and let M be an infinite unitary R-module. Call M homomorphically congruent (HC for short) provided M/N ? M for every submodule N of M for which |M/N| = |M|. In this article, we study HC modules over commutative rings. After a fairly comprehensive review of the literature, several natural examples are presented to motivate our study. We then prove some general results on HC modules, including HC module-theoretic characterizations of discrete valuation rings, almost Dedekind domains, and fields. We also provide a characterization of the HC modules over a Dedekind domain, extending Scott's classification over ? in [22 Scott , W. R. ( 1955 ). On infinite groups . Pacific J. Math. 5 : 589 – 598 .[Crossref] , [Google Scholar]]. Finally, we close with some open questions.     

On Goldie Extending Modules with Condition (C2)

A module M is said to be continuous if it is extending with the condition (C₂) (cf. [6 Mohamed, S. H., Müller, B. J. (1999). Continuous and Discrete Modules. London Math. Soc. LNS, Vol. 147. Cambridge: Cambridge Univ. Press. [Google Scholar]], [7 Oshiro, K. (1983). Continuous modules and quasi-continuous modules. Osaka J. Math. 20:681–694.[Web of Science ®] , [Google Scholar]]). In this article, we consider a 𝒢-extending module with (C₂) which is a generalization of a continuous module. First, we show that any 𝒢-extending module with (C₂) satisfies the exchange property. We also prove that, if M₁ and M₂ are 𝒢-extending modules with (C₂), then M₁ ⊕ M₂ is 𝒢-extending with (C₂) if and only if M_i is M_j-ejective (i ≠ j).     

On Some Artinian QF-3 Rings

Carl Faith in 2003 introduced and investigated an interesting class of rings over which every cyclic right module has Σ-injective injective hull (abbr., right CSI-rings) [5 Faith , C. ( 2003 ). When cyclic modules have Σ-injective hulls . Comm. Algebra 13 : 4161 – 4173 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. Inspired by this we investigate rings over which every cyclic right R-module has a projective Σ-injective injective hull. We show that a ring R satisfies this condition if and only if R is right artinian, the injective hull of R _R is projective and every simple right R-module is embedded in R _R . We also characterize right artinian rings in terms of injective faithful right ideals and right CSI-rings.     

A ring with FT-rank two

We construct a ring which admits a 2-generated, faithful torsion module but lacks a cyclic faithful torsion module. This answers a question by Oman and Schwiebert [1 Oman, G., Schwiebert, R. (2012). Rings which admit faithful torsion modules. Commun. Algebra 40(6):2184–2198.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar], 2 Oman, G., Schwiebert, R. (2012). Rings which admit faithful torsion modules II. J. Algebra Appl. 11(3):1250054 (12 p.).[Crossref], [Web of Science ®] , [Google Scholar]].     

Strongly Flat Modules over Matlis Domains

Strongly flat modules were introduced by Bazzoni–Salce [3 Bazzoni , S. , Salce , L. ( 2003 ). Almost perfect domains . Colloq. Math. 95 : 285 – 301 .[Crossref] , [Google Scholar]] and used to characterize almost perfect domains. Here we wish to study strongly flat modules, more generally, over Matlis domains; these are integral domains R such that the field of quotients Q has projective dimension 1. In Section 2, criteria are proved for strong flatness. We also prove that over arbitrary domains, strongly flat submodules of projective modules are projective (Theorem 3.2), in particular, strongly flat ideals are projective (Corollary 3.4) and use these results to show that the strongly flat dimension (which makes sense over Matlis domains) coincides with the projective dimension whenever it is > 1.     

Loewy Modules with Finite Loewy Invariants and Max Modules with Finite Radical Invariants

Over a commutative ring R, a module is artinian if and only if it is a Loewy module with finite Loewy invariants [5 Facchini , A. ( 1981 ). Loewy and artinian modules over commutative rings . Ann. Mat. Pura Appl. 128 : 359 – 374 .[Crossref], [Web of Science ®] , [Google Scholar]]. In this paper, we show that this is not necesarily true for modules over noncommutative rings R, though every artinian module is always a Loewy module with finite Loewy invariants. We prove that every Loewy module with finite Loewy invariants has a semilocal endomorphism ring, thus generalizing a result proved by Camps and Dicks for artinian modules [3 Camps , R. , Dicks , W. ( 1993 ). On semilocal rings . Israel J. Math. 81 : 203 – 211 .[Crossref], [Web of Science ®] , [Google Scholar]]. Finally, we obtain similar results for the dual class of max modules.     

On Semiprime Multiplication Modules over Pullback Rings

We classify all those indecomposable semiprime multiplication modules with finite-dimensional top over pullback of two Dedekind domains. We extend the definition and results given in [9 Ebrahimi Atani , S. , Farzalipour , F. ( 2009 ). Weak multiplication modules over a pullback of Dedekind domains . Colloquium Math. 114 : 99 – 112 .[Crossref] , [Google Scholar]] to a more general semiprime multiplication modules case.     

Irreducible Modules for the Lie Algebra of Divergence Zero Vector Fields on a Torus

This article investigates the irreducibility of certain representations for the Lie algebra of divergence zero vector fields on a torus. In [2 Eswara Rao, S. (1996). Irreducible representations of the Lie-algebra of the diffeomorphisms of a d-dimensional torus. J. Algebra 182(2):401–421.[Crossref], [Web of Science ®] , [Google Scholar]] Rao constructs modules for the Lie algebra of polynomial vector fields on an N-dimensional torus, and determines the conditions for irreducibility. The current article considers the restriction of these modules to the subalgebra of divergence zero vector fields. It is shown here that Rao's results transfer to similar irreducibility conditions for the Lie algebra of divergence zero vector fields.     

Gorenstein FP-Injective Dimension for Complexes

This article is a continuous work of [17 Hu , J. , Zhang , D. ( 2013 ). Weak AB-context for FP-injective modules with respect to semidualizing modules . J. Algebra Appl. 12 ( 7 ): 1350039 .[Crossref], [Web of Science ®] , [Google Scholar]], where the coauthors introduced the notion of 𝒢-FP-injective R-modules. In this article, we define a notion of 𝒢-FP-injective dimension for complexes over left coherent rings. To investigate the relationships between 𝒢-FP-injective dimension and FP-injective dimension for complexes, the complete cohomology group bases on FP-injectives is given.     

On H-supplemented modules over a right perfect ring

In 1971, Koehler [11 Koehler, A. (1971). Quasi-projective and quasi-injective modules. Pac. J. Math. 36(3):713–720.[Crossref], [Web of Science ®] , [Google Scholar]] proved a structure theorem for quasi-projective modules over right perfect rings by using results of Wu–Jans [22 Wu, L. E. T., Jans, J. P. (1967). On quasi-projectives. Illinois J. Math. 11:439–448. [Google Scholar]]. Later Mohamed–Singh [17 Mohamed, S. H., Singh, S. (1977). Generalizations of decomposition theorems known over perfect rings. J. Aust. Math. Soc. Ser. A 24(4):496–510.[Crossref] , [Google Scholar]] studied discrete modules over right perfect rings and gave decomposition theorems for these modules. Moreover, Oshiro [18 Oshiro, K. (1983). Semiperfect modules and quasi-semiperfect modules. Osaka J. Math. 20:337–372.[Web of Science ®] , [Google Scholar]] deeply studied (quasi-)discrete modules over general rings. In this paper, we consider that decomposition theorems for H-supplemented modules with the condition (D₂) or (D₃) over right perfect rings.     

A Tensor Product Factorization for Certain Tilting Modules

Let G be a semisimple, simply connected linear algebraic group over an algebraically closed field k of characteristic p > 0. In a recent article [6 Doty , S. R. ( 2009 ). Factoring tilting modules for algebraic groups . Journal of Lie Theory 19 ( 3 ): 531 – 535 .[Web of Science ®] , [Google Scholar]], Doty introduces the notion of r-minuscule weight and exhibits a tensor product factorization of a corresponding tilting module under the assumption p ≥ 2h ? 2, where h is the Coxeter number. We remove this restriction and consider some variations involving the more general notion of (p, r)-minuscule weights.