The purpose of this paper is to further the study of countably thick modules via weak injectivity. Among others, for some classes $ℳ$ of modules in $\sigma \left[M\right]$ we study when direct sums of modules from $ℳ$ satisfies a property $\mathbb{P}$ in $\sigma \left[M\right]$. In particular, we get characterization of locally countably thick modules, a generalization of locally q.f.d. modules.

In our recent paper (J. Algebra 345 (2011)) we prove that the deformed preprojective algebras of generalized Dynkin type ₙ (in the sense of our earlier work in Trans. Amer Math. Soc. 359 (2007)) are exactly (up to isomorphism) the stable Auslander algebras of simple plane singularities of Dynkin type ${}_{2n}$. In this article we complete the picture by showing that the deformed mesh algebras of Dynkin type ℂₙ are isomorphic to the canonical mesh algebras of type ℂₙ, and hence to the stable Auslander algebras...

We complete the derived equivalence classification of all weakly symmetric algebras of domestic type over an algebraically closed field, by solving the problem of distinguishing standard and nonstandard algebras up to stable equivalence, and hence derived equivalence. As a consequence, a complete stable equivalence classification of weakly symmetric algebras of domestic type is obtained.

Let $R⋉M$ be a trivial extension of a ring $R$ by an $R$-$R$-bimodule $M$ such that $M_R$, ${}_{R}M$, ${(R,0)}_{R⋉M}$ and ${}_{R⋉M}(R,0)$ have finite flat dimensions. We prove that $(X,\alpha )$ is a Ding projective left $R⋉M$-module if and only if the sequence $M{\otimes }_{R}M{\otimes }_{R}X\stackrel{M\otimes \alpha }{⟶}M{\otimes }_{R}X\stackrel{\alpha }{\to }X$ is exact and $\mathrm{coker}\left(\alpha \right)$ is a Ding projective left $R$-module. Analogously, we explicitly describe Ding injective $R⋉M$-modules. As applications, we characterize Ding projective and Ding injective modules over Morita context rings with zero bimodule homomorphisms.

Let $C$ be a semidualizing module over a commutative ring. We first investigate the properties of $C$-dual, $C$-torsionless and $C$-reflexive modules. Then we characterize some rings such as coherent rings, $\Pi$-coherent rings and FP-injectivity of $C$ using $C$-dual, $C$-torsionless and $C$-reflexive properties of some special modules.

In this paper we study a condition right FGTF on a ring R, namely when all finitely generated torsionless right R-modules embed in a free module. We show that for a von Neuman regular (VNR) ring R the condition is equivalent to every matrix ring Rn is a Baer ring; and this is right-left symmetric. Furthermore, for any Utumi VNR, this can be strengthened: R is FGTF iff R is self-injective.

In this article, we study modules with the weak $FI$-extending property. We prove that if $M$ satisfies weak $FI$-extending, pseudo duo, $C_3$ properties and $M/\mathrm{Soc}M$ has finite uniform dimension then $M$ decomposes into a direct sum of a semisimple submodule and a submodule of finite uniform dimension. In particular, if $M$ satisfies the weak $FI$-extending, pseudo duo, $C_3$ properties and ascending (or descending) chain condition on essential submodules then $M=M_1\oplus M_2$ for some semisimple submodule $M_1$ and Noetherian (or Artinian, respectively)...

An $R$-module $M$ is said to be an extending module if every closed submodule of $M$ is a direct summand. In this paper we introduce and investigate the concept of a type 2 $\tau$-extending module, where $\tau$ is a hereditary torsion theory on $\text{Mod}$-$R$. An $R$-module $M$ is called type 2 $\tau$-extending if every type 2 $\tau$-closed submodule of $M$ is a direct summand of $M$. If ${\tau }_I$ is the torsion theory on $\text{Mod}$-$R$ corresponding to an idempotent ideal $I$ of $R$ and $M$ is a type 2 ${\tau }_I$-extending $R$-module, then the question of whether or not $M/MI$ is...