In the first part, we study algebras A such that A = R ⨿ I, where R is a subalgebra and I a two-sided nilpotent ideal. Under certain conditions on I, we show that A is standardly stratified if and only if R is standardly stratified. Next, for $A=\left[\begin{array}{cc}U& 0\\ M& V\end{array}\right]$, we show that A is standardly stratified if and only if the algebra R = U × V is standardly stratified and ${}_{V}M$ is a good V-module.

The class of n-fundamental algebras is introduced. It is a subclass of string algebras. For n-fundamental algebras we study the problem of when the Auslander-Reiten quiver contains, at the beginning or at the end, a component which is not generalized standard.

Let $A$ be a standard Koszul standardly stratified algebra and $X$ an $A$-module. The paper investigates conditions which imply that the module ${\mathrm{Ext}}_A^{*}\left(X\right)$ over the Yoneda extension algebra $A^{*}$ is filtered by standard modules. In particular, we prove that the Yoneda extension algebra of $A$ is also standardly stratified. This is a generalization of similar results on quasi-hereditary and on graded standardly stratified algebras.

In this paper, we prove that any pure submodule of a strict Mittag-Leffler module is a locally split submodule. As applications, we discuss some relations between locally split monomorphisms and locally split epimorphisms and give a partial answer to the open problem whether Gorenstein projective modules are Ding projective.

Let Λ be an artinian ring and let 𝔯 denote its Jacobson radical. We show that a simple module of finite projective dimension has no self-extensions when Λ is graded by its radical, with at most two simple modules and 𝔯⁴ = 0, in particular, when Λ is a finite-dimensional algebra over an algebraically closed field with at most two simple modules and 𝔯³ = 0.

An exchange ring $R$ is strongly separative provided that for all finitely generated projective right $R$-modules $A$ and $B$, $A\oplus A\cong A\oplus B\Rightarrow A\cong B$. We prove that an exchange ring $R$ is strongly separative if and only if for any corner $S$ of $R$, $aS+bS=S$ implies that there exist $u,v\in S$ such that $au=bv$ and $Su+Sv=S$ if and only if for any corner $S$ of $R$, $aS+bS=S$ implies that there exists a right invertible matrix $\left(\begin{array}{cc}a& b\\ & *\end{array}\right)\in M_2\left(S\right)$. The dual assertions are also proved.

A $*$-ring $R$ is strongly 2-nil-$*$-clean if every element in $R$ is the sum of two projections and a nilpotent that commute. Fundamental properties of such $*$-rings are obtained. We prove that a $*$-ring $R$ is strongly 2-nil-$*$-clean if and only if for all $a\in R$, $a^2\in R$ is strongly nil-$*$-clean, if and only if for any $a\in R$ there exists a $*$-tripotent $e\in R$ such that $a-e\in R$ is nilpotent and $ea=ae$, if and only if $R$ is a strongly $*$-clean SN ring, if and only if $R$ is abelian, $J\left(R\right)$ is nil and $R/J\left(R\right)$ is $*$-tripotent. Furthermore, we explore the structure...

Let $\Lambda =\left(\begin{array}{cc}A& M\\ 0& B\end{array}\right)$ be an Artin algebra. In view of the characterization of finitely generated Gorenstein injective $\Lambda$-modules under the condition that $M$ is a cocompatible $(A,B)$-bimodule, we establish a recollement of the stable category $\overline{\mathrm{Ginj}\left(\Lambda \right)}$. We also determine all strongly complete injective resolutions and all strongly Gorenstein injective modules over $\Lambda$.

Let $𝒲$ be a self-orthogonal class of left $R$-modules. We introduce a class of modules, which is called strongly $𝒲$-Gorenstein modules, and give some equivalent characterizations of them. Many important classes of modules are included in these modules. It is proved that the class of strongly $𝒲$-Gorenstein modules is closed under finite direct sums. We also give some sufficient conditions under which the property of strongly $𝒲$-Gorenstein module can be inherited by its submodules and quotient modules....

An associated ring R with identity is said to be a left FTF ring when the class of the submodules of flat left R-modules is closed under injective hulls and direct products. We prove (Theorem 3.5) that a strongly graded ring R by a locally finite group G is FTF if and only if Re is left FTF, where e is a neutral element of G. This provides new examples of left FTF rings. Some consequences of this Theorem are given.

We interpret the collection of invertible bimodules as a groupoid and call it the Picard groupoid. We use this groupoid to generalize the classical construction of crossed products to what we call groupoid crossed products, and show that these coincide with the class of strongly groupoid graded rings. We then use groupoid crossed products to obtain a generalization from the group graded situation to the groupoid graded case of the bijection from a second cohomology group, defined by the grading...

Let $𝒯$ be a weak torsion class of left $R$-modules and $n$ a positive integer. A left $R$-module $M$ is called $(𝒯,n)$-injective if ${\mathrm{Ext}}_R^n(C,M)=0$ for each $(𝒯,n+1)$-presented left $R$-module $C$; a right $R$-module $M$ is called $(𝒯,n)$-flat if ${\mathrm{Tor}}_n^R(M,C)=0$ for each $(𝒯,n+1)$-presented left $R$-module $C$; a left $R$-module $M$ is called $(𝒯,n)$-projective if ${\mathrm{Ext}}_R^n(M,N)=0$ for each $(𝒯,n)$-injective left $R$-module $N$; the ring $R$ is called strongly $(𝒯,n)$-coherent if whenever $0\to K\to P\to C\to 0$ is exact, where $C$ is $(𝒯,n+1)$-presented and $P$ is finitely generated projective, then $K$ is $(𝒯,n)$-projective; the ring $R$ is called $(𝒯,n)$-semihereditary...