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...

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...