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 the almost Gorenstein property of determinantal rings

In this paper, we investigate the question of when the determinantal ring R over a field k is an almost Gorenstein local/graded ring in the sense of [14 Goto, S., Takahashi, R., Taniguchi, N. (2015). Almost Gorenstein rings - towards a theory of higher dimension. J. Pure Appl. Algebra 219:2666–2712.[Crossref], [Web of Science ®] , [Google Scholar]]. As a consequence of the main result, we see that if R is a non-Gorenstein almost Gorenstein local/graded ring, then the ring R has a minimal multiplicity.     

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.     

Strongly J-Clean Matrices Over Local Rings

An element of a ring is called strongly J-clean provided that it can be written as the sum of an idempotent and an element in its Jacobson radical that commute. We investigate, in this article, a single strongly J-clean 2 × 2 matrix over a noncommutative local ring. The criteria on strong J-cleanness of 2 × 2 matrices in terms of a quadratic equation are given. These extend the corresponding results in [8 Li , B. ( 2009 ). Strongly clean matrix rings over noncommutative local rings . Bull. Korean Math. Soc. 46 : 71 – 78 .[Crossref], [Web of Science ®] , [Google Scholar], Theorems 2.7 and 3.2], [9 Li , Y. ( 2007 ). Strongly clean matrix rings over local rings . J. Algebra 312 : 397 – 404 .[Crossref], [Web of Science ®] , [Google Scholar], Theorem 2.6], and [11 Yang , X. , Zhou , Y. ( 2008 ). Strongly cleanness of the 2 × 2 matrix ring over a general local ring . J. Algebra 320 : 2280 – 2290 .[Crossref], [Web of Science ®] , [Google Scholar], Theorem 7].     

Auslander–Reiten Triangles in Homotopy Categories

Let A be an artin algebra. We show that the bounded homotopy category of finitely generated right A-modules has Auslander–Reiten triangles. Two applications are given: (1) we provide an alternative proof of a theorem of Happel in [14 Happel, D. (1988). Triangulated Categories in the Representation Theory of Finite-dimensional Algebras. London Mathematical Society Lecture Note Series, vol. 119. Cambridge: Cambridge University Press.[Crossref] , [Google Scholar]]; (2) we prove that over a Gorenstein algebra, the bounded homotopy category of finitely generated Gorenstein projective (resp. injective) modules, admits Auslander–Reiten triangles, which improve a main result in [12 Nan, G. (2012). Auslander-Reiten triangles on Gorenstein derived categories. Comm. Algebra 40:3912–3919.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]].     

Gorenstein coresolving categories

Let 𝒜 be an abelian category. A subcategory 𝒳 of 𝒜 is called coresolving if 𝒳 is closed under extensions and cokernels of monomorphisms and contains all injective objects of 𝒜. In this paper, we introduce and study Gorenstein coresolving categories, which unify the following notions: Gorenstein injective modules [8 Enochs, E. E., Jenda, O. M. G. (1995). Gorenstein injective and projective modules. Math. Z. 220:611–633.[Crossref], [Web of Science ®] , [Google Scholar]], Gorenstein FP-injective modules [20 Mao, L. X., Ding, N. Q. (2008). Gorenstein FP-injective and Gorenstein flat modules. J. Algebra Appl. 7:491–506.[Crossref], [Web of Science ®] , [Google Scholar]], Gorenstein AC-injective modules [3 Bravo, D., Gillespie, J. (2016). Absolutely clean, level, and Gorenstein AC-injective complexes. Commun. Algebra 44:2213–2233.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]], and so on. Then we define a resolution dimension relative to the Gorenstein coresolving category 𝒢?_𝒳(𝒜). We investigate the properties of the homological dimension and unify some important properties possessed by some known homological dimensions. In addition, we study stability of the Gorenstein coresolving category 𝒢?_𝒳(𝒜) and apply the obtained properties to special subcategories and in particular to module categories.     

On Strongly Clean Matrix and Triangular Matrix Rings

A ring R with identity is called “clean” if every element of R is the sum of an idempotent and a unit, and R is called “strongly clean” if every element of R is the sum of an idempotent and a unit that commute. Strongly clean rings are “additive analogs” of strongly regular rings, where a ring R is strongly regular if every element of R is the product of an idempotent and a unit that commute. Strongly clean rings were introduced in Nicholson (1999 Nicholson , W. K. (1999). Strongly clean rings and Fitting's lemma. Comm. Algebra 27:3583–3592. [CSA] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) where their connection with strongly π-regular rings and hence to Fitting's Lemma were discussed. Local rings and strongly π-regular rings are all strongly clean. In this article, we identify new families of strongly clean rings through matrix rings and triangular matrix rings. For instance, it is proven that the 2 × 2 matrix ring over the ring of p-adic integers and the triangular matrix ring over a commutative semiperfect ring are all strongly clean.     

Corrigendum To: Goldie Extending Modules

This corrigendum is written to correct the proof of Theorem 5.3 of 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]].     

Corrigendum to: Goldie Extending Modules

This corrigendum is written to correct an error in Corollary 2.5(ii) and an error in the proof of the converse of Theorem 2.7 of Akalan, Birkenmeier, and Tercan [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]].     

A Question on Strongly Clean Rings

A new characterization of a strongly clean ring is given. And it is proven that if R is a strongly clean ring, then eRe is a strongly clean ring for e ² = e ∈ R, which answers a question of Nicholson (1999 Nicholson , W. K. ( 1999 ). Strongly clean rings and Fitting's lemma . Comm. Algebra 27 ( 8 ): 3583 – 3592 . [CSA] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) in the affirmative.     

A Class of Gorenstein Artin Algebras of Embedding Dimension Four

In this article, we study height four graded Gorenstein ideals I in k[x, y, z, w] such that I ₂ is of height one and generated by three quadrics. After a suitable linear change of variables, I ∩ k[x, y, z] is either Gorenstein or of type two. The former case was studied by Iarrobino and Srinivasan [8 Iarrobino , A. , Srinivasan , H. ( 2005 ). Artininan Gorenstein algebras of embedding dimension four: components of ? Gor (H) for H = (1, 4, 7,…, 1) . Journal of Pure and Applied Algebra 201 : 62 – 96 .[Crossref], [Web of Science ®] , [Google Scholar]] where they give the structure of the ideal and its resolution. We study the latter case and give the structure of these ideals and their minimal resolution. We also explicitly write the form of the generators of I and the maps in the free resolution of R/I.     

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.     

On Serial Noetherian Rings

It is well known that every serial Noetherian ring satisfies the restricted minimum condition. In particular, following Warfield (1975 Warfield , R. B. ( 1975 ). Serial rings and finitely presented modules . J. Algebra 37 : 187 – 222 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]), such a ring is a direct sum of an Artinian ring and hereditary prime rings. The aim of this note is to show that every serial ring having the restricted minimum condition is Noetherian.     

A Note on Strongly Clean Matrix Rings

Let R be an associative ring with identity. An element a ∈ R is called strongly clean if a = e + u with e ² = e ∈ R, u a unit of R, and eu = ue. A ring R is called strongly clean if every element of R is strongly clean. Strongly clean rings were introduced by Nicholson [7 Nicholson , W. K. ( 1999 ). Strongly clean rings and Fitting's lemma . Comm. Algebra 27 : 3583 – 3592 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. It is unknown yet when a matrix ring over a strongly clean ring is strongly clean. Several articles discussed this topic when R is local or strongly π-regular. In this note, necessary conditions for the matrix ring 𝕄 _n (R) (n > 1) over an arbitrary ring R to be strongly clean are given, and the strongly clean property of 𝕄₂(RC ₂) over the group ring RC ₂ with R local is obtained.     

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]].     

Almost Dual Pairs and Definable Classes of Modules

In [7 Holm , H. ( 2010 ). Modules with cosupport and injective functors . Algebr. Represent. Theor. 13 : 543 – 560 .[Crossref], [Web of Science ®] , [Google Scholar]] Holm considers categories of right modules dual to those with support in a set of finitely presented modules. We extend some of his results by placing them in the context of elementary duality on definable subcategories. In doing so we also prove that dual modules have enough indecomposable direct summands.     

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.     

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.