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....

Let $R$ be a left Noetherian ring, $S$ a right Noetherian ring and ${}_R\omega$ a Wakamatsu tilting module with $S=\mathrm{End}{(}_R\omega )$. We introduce the notion of the $\omega$-torsionfree dimension of finitely generated $R$-modules and give some criteria for computing it. For any $n\ge 0$, we prove that ${\mathrm{l}.\mathrm{id}}_R\left(\omega \right)={\mathrm{r}.\mathrm{id}}_S\left(\omega \right)\le n$ if and only if every finitely generated left $R$-module and every finitely generated right $S$-module have $\omega$-torsionfree dimension at most $n$, if and only if every finitely generated left $R$-module (or right $S$-module) has generalized Gorenstein dimension...

We study the weak dimension of a group-graded ring using methods developed in [B1], [Q] and [R]. We prove that if R is a G-graded ring with G locally finite and the order of every subgroup of G is invertible in R, then the graded weak dimension of R is equal to the ungraded one.

Let $K$ be a field, and let $G$ be a group. In the present paper, we investigate when the group ring $K\left[G\right]$ has finite weak dimension and finite Gorenstein weak dimension. We give some analogous versions of Serre’s theorem for the weak dimension and the Gorenstein weak dimension.

We introduce and study the concepts of weak $n$-injective and weak $n$-flat modules in terms of super finitely presented modules whose projective dimension is at most $n$, which generalize the $n$-FP-injective and $n$-flat modules. We show that the class of all weak $n$-injective $R$-modules is injectively resolving, whereas that of weak $n$-flat right $R$-modules is projectively resolving and the class of weak $n$-injective (or weak $n$-flat) modules together with its left (or right) orthogonal class forms a hereditary...

Using a lattice-theoretical approach we find characterizations of modules with finite uniform dimension and of modules with finite hollow dimension.