In this note we show that for a ${*}^n$-module, in particular, an almost $n$-tilting module, $P$ over a ring $R$ with $A={\mathrm{E}nd}_{R}P$ such that $P_A$ has finite flat dimension, the upper bound of the global dimension of $A$ can be estimated by the global dimension of $R$ and hence generalize the corresponding results in tilting theory and the ones in the theory of $*$-modules. As an application, we show that for a finitely generated projective module over a VN regular ring $R$, the global dimension of its endomorphism ring is not more...

Let $R$ be a ring. In two previous articles [12, 14] we studied the homotopy category $𝐊\left(R\text{-}\mathrm{Proj}\right)$ of projective $R$-modules. We produced a set of generators for this category, proved that the category is ${\aleph }_1$-compactly generated for any ring $R$, and showed that it need not always be compactly generated, but is for sufficiently nice $R$. We furthermore analyzed the inclusion $j_{!}^{}:𝐊\left(R\text{-}\mathrm{Proj}\right)\to 𝐊\left(R\text{-}\mathrm{Flat}\right)$ and the orthogonal subcategory $𝒮={𝐊\left(R\text{-}\mathrm{Proj}\right)}^{\perp }$. And we even showed that the inclusion $𝒮\to 𝐊\left(R\text{-}\mathrm{Flat}\right)$ has a right adjoint; this forces some natural map to be an equivalence...

Using derived categories, we develop an alternative approach to defining Koszulness for positively graded algebras where the degree zero part is not necessarily semisimple.

An $R$-module $M$ is said to be an extending module if every closed submodule of $M$ is a direct summand. In this paper we introduce and investigate the concept of a type 2 $\tau$-extending module, where $\tau$ is a hereditary torsion theory on $\text{Mod}$-$R$. An $R$-module $M$ is called type 2 $\tau$-extending if every type 2 $\tau$-closed submodule of $M$ is a direct summand of $M$. If ${\tau }_I$ is the torsion theory on $\text{Mod}$-$R$ corresponding to an idempotent ideal $I$ of $R$ and $M$ is a type 2 ${\tau }_I$-extending $R$-module, then the question of whether or not $M/MI$ is...