In this note, for a ring endomorphism $\alpha$ and an $\alpha$-derivation $\delta$ of a ring $R$, the notion of weakened $(\alpha ,\delta )$-skew Armendariz rings is introduced as a generalization of $\alpha$-rigid rings and weak Armendariz rings. It is proved that $R$ is a weakened $(\alpha ,\delta )$-skew Armendariz ring if and only if $T_n\left(R\right)$ is weakened $(\overline{\alpha },\overline{\delta })$-skew Armendariz if and only if $R\left[x\right]/\left(x^n\right)$ is weakened $(\overline{\alpha },\overline{\delta })$-skew Armendariz ring for any positive integer $n$.

The purpose of this paper is to further the study of weakly injective and weakly projective modules as a generalization of injective and projective modules. For a locally q.f.d. module $M$, there exists a module $K\in \sigma \left[M\right]$ such that $K\oplus N$ is weakly injective in $\sigma \left[M\right]$, for any $N\in \sigma \left[M\right]$. Similarly, if $M$ is projective and right perfect in $\sigma \left[M\right]$, then there exists a module $K\in \sigma \left[M\right]$ such that $K\oplus N$ is weakly projective in $\sigma \left[M\right]$, for any $N\in \sigma \left[M\right]$. Consequently, over a right perfect ring every module is a direct summand of a weakly projective module. For...

We give necessary and sufficient conditions for a wing of an Auslander-Reiten quiver of a selfinjective algebra to be the wing of the radical of an indecomposable projective module. Moreover, a characterization of indecomposable Nakayama algebras of Loewy length ≥ 3 is obtained.

Let $M$ be a module and $\mu$ be a class of modules in $Mod-R$ which is closed under isomorphisms and submodules. As a generalization of essential submodules Özcan in [8] defines a $\mu$-essential submodule provided it has a non-zero intersection with any non-zero submodule in $\mu$. We define and investigate $\mu$-singular modules. We also introduce $\mu$-extending and weakly $\mu$-extending modules and mainly study weakly $\mu$-extending modules. We give some characterizations of $\mu$-co-H-rings by weakly $\mu$-extending modules. Let $R$...

Let $R$ be a 2-torsion free prime ring and let $\sigma ,\tau$ be automorphisms of $R$. For any $x,y\in R$, set ${[x,y]}_{\sigma ,\tau }=x\sigma \left(y\right)-\tau \left(y\right)x$. Suppose that $d$ is a $(\sigma ,\tau )$-derivation defined on $R$. In the present paper it is shown that $\left(i\right)$ if $R$ satisfies ${[d\left(x\right),x]}_{\sigma ,\tau }=0$, then either $d=0$ or $R$ is commutative $\left(ii\right)$ if $I$ is a nonzero ideal of $R$ such that $\left[d\right(x),d(y\left)\right]=0$, for all $x,y\in I$, and $d$ commutes with both $\sigma$ and $\tau$, then either $d=0$ or $R$ is commutative. $\left(iii\right)$ if $I$ is a nonzero ideal of $R$ such that $d\left(xy\right)=d\left(yx\right)$, for all $x,y\in I$, and $d$ commutes with $\tau$, then $R$ is commutative. Finally a related result has been obtain for $(\sigma ,\tau )$-derivation....

In this paper we introduce the concept of $\tau$-extending modules by $\tau$-rational submodules and study some properties of such modules. It is shown that the set of all $\tau$-rational left ideals of ${}_{R}R$ is a Gabriel filter. An $R$-module $M$ is called $\tau$-extending if every submodule of $M$ is $\tau$-rational in a direct summand of $M$. It is proved that $M$ is $\tau$-extending if and only if $M=Re{j}_{M}E(R/\tau \left(R\right))\oplus N$, such that $N$ is a $\tau$-extending submodule of $M$. An example is given to show that the direct sum of $\tau$-extending modules need not be $\tau$-extending....

The structure of filtered algebras of Grothendieck's differential operators on a smooth fat point in a curve and graded Poisson algebras of their principal symbols is explicitly determined. A related infinitesimal-birational duality realized by a Springer type resolution of singularities and the Fourier transformation is presented. This algebro-geometrical duality is quantized in appropriate sense and its quantum origin is explained.

We classify one-directed indecomposable pure injective modules over finite-dimensional string algebras.

We introduce the right (left) Gorenstein subcategory relative to an additive subcategory $𝒞$ of an abelian category $𝒜$, and prove that the right Gorenstein subcategory $r𝒢\left(𝒞\right)$ is closed under extensions, kernels of epimorphisms, direct summands and finite direct sums. When $𝒞$ is self-orthogonal, we give a characterization for objects in $r𝒢\left(𝒞\right)$, and prove that any object in $𝒜$ with finite $r𝒢\left(𝒞\right)$-projective dimension is isomorphic to a kernel (or a cokernel) of a morphism from an object in $𝒜$ with finite $𝒞$-projective dimension...