Let $A$ and $B$ be rings, $U$ a $(B, A)$-bimodule and $T=left(begin{smallmatrix} A & 0 U & B end{smallmatrix}right)$ be the triangular matrix ring. In this paper, we characterize the Gorenstein homological dimensions of modules over $T$, and discuss
when a left $T$-module is strongly Gorenstein projective or strongly Gorenstein injective module.
The notion of an $mathcal{L}$ complex (for a given class of $R$-modules $mathcal{L}$) was introduced by Gillespie: a complex $C$ is called $mathcal{L}$ complex if $C$ is exact and $Z_{i}(C)$ is in $mathcal{L}$ for all $iin mathbb{Z}$. Let $widetilde{
mathcal{L}}$ stand for the class of all $mathcal{L}$ complexes. In this paper, we give sufficient condition on a class of $R$-modules such that every complex has a special $widetilde{mathcal{L}}$-precover (resp., $widetilde{mathcal{L}}$-preenvelope). As applications, we obtain that every complex has a special projective precover and a special injective preenvelope, over a coherent ring every complex has a special FP-injective preenvelope, and over a noetherian ring every complex has a special $widetilde{mathcal{GI}}$-preenvelope, where $mathcal{GI}$ denotes the class of Gorenstein injective modules.
A left $R$-module $M$ is called two-degree Ding projective if there exists an exact sequence $...longrightarrow D_{1}longrightarrow D_{0}longrightarrow D_{-1}longrightarrow D_{-2}longrightarrow...$ of Ding projective left $R$-modules such that $Mcong
ker (D_{0}longrightarrow D_{-1})$ and $Hom_{R} (-, F)$ leaves the sequence exact for any flat (or Gorenstein flat) left $R$-module $F$. In this paper, we show that the two-degree Ding projective modules are nothing more than the Ding projective modules.
In this paper, we define and study a notion of Ding projective dimension for complexes of left modules over associative rings. In particular, we consider the class of homologically bounded below complexes of left R-modules, and show that Ding projective dimension has a nice functorial description.
