On graded nil clean rings

In this paper we introduce and study the notion of a graded (strongly) nil clean ring which is group graded. We also deal with extensions of graded (strongly) nil clean rings to graded matrix rings and to graded group rings. The question of when nil cleanness of the component, which corresponds to the neutral element of a group, implies graded nil cleanness of the whole graded ring is examined. Similar question is discussed in the case of groupoid graded rings as well.     

Noetherian rings with almost injective simple modules

We characterize right Noetherian rings over which all simple modules are almost injective. It is proved that R is such a ring, if and only if, the complements of semisimple submodules of every R-module M are direct summands of M, if and only if, R is a finite direct sum of right ideals I_r, where I_r is either a Noetherian V-module with zero socle, or a simple module, or an injective module of length 2. A commutative Noetherian ring for which all simple modules are almost injective is precisely a finite direct product of rings R_i, where R_i is either a field or a quasi-Frobenius ring of length 2. We show that for commutative rings whose all simple modules are almost injective, the properties of Kasch, (semi)perfect, semilocal, quasi-Frobenius, Artinian, and Noetherian coincide.     

On the completeness of factor rings

Let be a complete local domain containing the integers with maximal ideal such that is at least the cardinality of the real numbers. Let be a nonmaximal prime ideal of such that is a regular local ring. We construct an excellent local ring such that the completion of is , the generic formal fiber of is local with maximal ideal and if is a nonzero ideal of , then is complete.

     

On extensions of matrix rings with skew Hochschild 2-cocycles

We study structures of Hochschild 2-cocycles related to endomorphisms and introduce a skew Hochschild 2-cocycle. We moreover define skew Hochschild extensions equipped with skew Hochschild 2-cocycles, and then we examine uniquely clean, Abelian, directly finite, symmetric, and reversible ring properties of skew Hochschild extensions and related ring systems. The results obtained here provide various kinds of examples of such rings. Especially, we give an answer negatively to the question of H. Lin and C. Xi for the corresponding Hochschild extensions of reversible (or semicommutative) rings. Finally, we establish three kinds of Hochschild extensions with Hochschild 2-cocycles and skew Hochschild 2-cocycles.     

PC—环与几乎优越扩张

本文我们讨论了这种环R的结构,R是凝聚环且R作为R－模是P－内射,我们称此环为PC－环,并证明了在几乎优越扩张下的不变性。     

On a class of coherent regular rings

The paper investigates a special class of quasi-local rings. It is shown that these rings are coherent and regular in the sense that every finitely generated submodule of a free module has a finite free resolution.

     

Serial rings and subdirect products

A basic Artinian serial ring can be realized as the subdirect product of factor rings of (S, M)-upper triangular matrix rings with S a local Artinian ring and M the maximal ideal of S. As an application the serial subdirect product of (S, M)-rings is shown to have self-duality.     

Descent of the canonical module in rings with the approximation property

Let be a local Noetherian Cohen-Macaulay ring with the approximation property. We show that admits a canonical module.

     

On graded Thierrin radicals of graded rings

In this paper, we study the graded Thierrin radical and the classical Thierrin radical of a graded ring, which is the direct sum of a family of its additive subgroups indexed by a nonempty set, under the assumption that the product of homogeneous elements is again homogeneous. There are two versions of this graded radical, the graded Thierrin and the large graded Thierrin radical. We establish several characterizations of the graded Thierrin radical and prove that the largest homogeneous ideal contained in the classical Thierrin radical of a graded ring coincides with the large graded Thierrin radical of that ring.     

On (Strongly) Gorenstein Von Neumann Regular Rings

In this article, we introduce and study the rings over which every module is (strongly) Gorenstein flat, which we call them (strongly) Gorenstein Von Neumann regular rings.     

一类广义遗传环

称环R为左亚遗传环，如果内射左R－模的商模是FG－内射的，给出了左亚遗传环的一些刻划，给出了左亚遗传环的半单环的条件，并研究了左亚遗传环的一些性质。     

Commutative weakly nil-neat group rings

Abstract

Let R be a ring and let G be a group. We prove a rather curious necessary and sufficient condition for the commutative group ring RG to be weakly nil-neat only in terms of R,G and their sections. This somewhat expands three recent results, namely those established by McGovern et al. in (J. Algebra Appl., 2015), by Danchev-McGovern in (J. Algebra, 2015) and by the present authors in (J. Math., Tokushima Univ., 2019), related to commutative nil-clean, weakly nil-clean and nil-neat group rings, respectively.     

On the equivalence of codes over rings and modules

In light of the result by Wood that codes over every finite Frobenius ring satisfy a version of the MacWilliams equivalence theorem, a proof for the converse is considered. A strategy is proposed that would reduce the question to problems dealing only with matrices over finite fields. Using this strategy, it is shown, among other things, that any left MacWilliams basic ring is Frobenius. The results and techniques in the paper also apply to related problems dealing with codes over modules.     

On arithmetic Macaulayfication of Noetherian rings

The Rees algebra is the homogeneous coordinate ring of a blowing-up. The present paper gives a necessary and sufficient condition for a Noetherian local ring to have a Cohen-Macaulay Rees algebra: A Noetherian local ring has a Cohen-Macaulay Rees algebra if and only if it is unmixed and all the formal fibers of it are Cohen-Macaulay. As a consequence of it, we characterize a homomorphic image of a Cohen-Macaulay local ring. For non-local rings, this paper gives only a sufficient condition. By using it, however, we obtain the affirmative answer to Sharp's conjecture. That is, a Noetherian ring having a dualizing complex is a homomorphic image of a finite-dimensional Gorenstein ring.

     

A note on quasi-Frobenius rings

It is shown that a semiperfect ring is quasi-Frobenius if and only if every closed submodule of is non-small, where denotes the direct sum of copies of the right -module and is the first infinite ordinal.

     

On GC2 modules and their endomorphism rings

Let R be a ring. A module M_R is said to be GC₂ if for any N≤ M with N? M, N is a direct summand of M. In this article, we give some characterizations and properties of GC₂ modules and their endomorphism rings, and many results on C ₂ modules and GC₂ rings are generalized to GC₂ modules.