Given a semiperfect two-sided noetherian ring Λ, we study two subcategories ${}_k\left(\Lambda \right)=M\in mod\Lambda |Ex{t}_{\Lambda }^j(TrM,\Lambda )=0\left(1\le j\le k\right)$ and ${}_k\left(\Lambda \right)=N\in mod\Lambda |Ex{t}_{\Lambda }^j(N,\Lambda )=0\left(1\le j\le k\right)$ of the category mod Λ of finitely generated right Λ-modules, where Tr M is Auslander’s transpose of M. In particular, we give another convenient description of the categories ${}_k\left(\Lambda \right)$ and ${}_k\left(\Lambda \right)$, and we study category equivalences and stable equivalences between them. Several results proved in [J. Algebra 301 (2006), 748-780] are extended to the case when Λ is a two-sided noetherian semiperfect ring.

Given an object in a category, we study its orbit algebra with respect to an endofunctor. We show that if the object is periodic, then its orbit algebra modulo nilpotence is a polynomial ring in one variable. This specializes to a result on Ext-algebras of periodic modules over Gorenstein algebras. We also obtain a criterion for an algebra to be of wild representation type.

Let $$𝕂$$ be an algebraically closed field. Consider a finite dimensional monomial relations algebra $$\Lambda =𝕂\Gamma /\phantom{𝕂\Gamma I}I$$ of finite global dimension, where Γ is a quiver and I an admissible ideal generated by a set of paths from the path algebra $$𝕂\Gamma$$ . There are many modules over Λ which may be represented graphically by a tree with respect to a top element, of which the indecomposable projectives are the most natural example. These trees possess branches which correspond to right subpaths of cycles in the quiver. A pattern...

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.

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....

We show that there is a one-to-one correspondence between basic cotilting complexes and certain contravariantly finite subcategories of the bounded derived category of an artin algebra. This is a triangulated version of a result by Auslander and Reiten. We use this to find an existence criterion for complements to exceptional complexes.

In this paper we compute injective, projective and flat dimensions of inverse polynomial modules as $R\left[x\right]$-modules. We also generalize Hom and Ext functors of inverse polynomial modules to any submonoid but we show Tor functor of inverse polynomial modules can be generalized only for a symmetric submonoid.

Let $S$ and $R$ be two associative rings, let ${}_S{C}_R$ be a semidualizing $(S,R)$-bimodule. We introduce and investigate properties of the totally reflexive module with respect to ${}_S{C}_R$ and we give a characterization of the class of the totally $C_R$-reflexive modules over any ring $R$. Moreover, we show that the totally $C_R$-reflexive module with finite projective dimension is exactly the finitely generated projective right $R$-module. We then study the relations between the class of totally reflexive modules and the Bass class...

The notion of Gorenstein rings in the commutative ring theory is generalized to that of Noetherian algebras which are not necessarily commutative. We faithfully follow in the steps of the commutative case: Gorenstein algebras will be defined using the notion of Cousin complexes developed by R. Y. Sharp [Sh1]. One of the goals of the present paper is the characterization of Gorenstein algebras in terms of Bass numbers. The commutative theory of Bass numbers turns out to carry over with no extra changes....