arXiv subjects
Publications and source records attributed to Lixin Mao.
Let $R\ltimes_{n}M$ be the $n$-trivial extension of a ring $R$ by an $R$-bimodule $M$. We first characterize projective, injective, flat and finitely generated modules over $R\ltimes_{n}M$. Then we describe when the ring $R\ltimes_{n}M$ is left perfect (Noetherian, coherent, Artinian, hereditary, Kasch). Finally, we establish some homological formulas of $n$-trivial extensions of rings.
Let $R$ be a commutative ring. An $R$-module $M$ is said to be $w$-split if Ext$_{R}^1(M,N)$ is a GV-torsion $R$-module for all $R$-modules $N$. It is known that every projective module is $w$-split, but the converse is not true in general. In this paper, we study the w-split dimension of a flat module. To do so, we introduce and study the so-called $w$-copure (resp., strongly $w$-copure) projective modules which is in some way a generalization of the notion of copure (resp., strongly copure) projective modules. An $R$-module $M$ is said to be $w$-copure projective (resp., strongly $w$-copure projective) if Ext$_{R}^1(M,N)$ (resp., Ext$_{R}^n(M,N)$) is a GV-torsion $R$-module for all flat $R$-modules $N$ and any $n\geq1$.
We introduce the concepts of generalized compatible and cocompatible bimodules in order to characterize Gorenstein projective, injective and flat modules over trivial ring extensions. Let $R\ltimes M$ be a trivial extension of a ring $R$ by an $R$-$R$-bimodule $M$ such that $M$ is a generalized compatible $R$-$R$-bimodule and $\textbf{Z}(R)$ is a generalized compatible $R\ltimes M$-$R\ltimes M$-bimodule. We prove that $(X,α)$ is a Gorenstein projective left $R\ltimes M$-module if and only if the sequence $M\otimes_R M\otimes_R X\stackrel{M\otimesα}\rightarrow M\otimes_R X\stackrelα\rightarrow X$ is exact and coker$(α)$ is a Gorenstein projective left $R$-module. Analogously, we explicitly characterize Gorenstein injective and flat modules over trivial ring extensions. As an application, we describe Gorenstein projective, injective and flat modules over Morita context rings with zero bimodule homomorphisms.
In Enochs' relative homological dimension theory occur the so called (co)resolvent and (co)proper dimensions which are defined using proper and coproper resolutions constructed by precovers and preenvelopes, respectively. Recently, some authors have been interested in relative homological dimensions defined by just exact sequences. In this paper, we contribute to the investigation of these relative homological dimensions. We first study the relation between these two kinds of relative homological dimensions and establish some "transfer results" under adjoint pairs. Then, relative global dimensions are studied which lead to nice characterizations of some properties of particular cases of self-orthogonal subcategories. At the end of the paper, relative derived functors are studied and generalizations of some known results of balance for relative homology are established.
Let $T=\biggl(\begin{matrix} A&0\\ U&B \end{matrix}\biggr)$ be a formal triangular matrix ring, where $A$ and $B$ are rings and $U$ is a $(B, A)$-bimodule. We prove that: (1) If $U_A$ and $_B U$ have finite flat dimensions, then a left $T$-module $\biggl(\begin{matrix} M_1\\ M_2\end{matrix}\biggr)_{φ^M}$ is Ding projective if and only if $M_1$ and $M_2/{\rm im}(φ^M)$ are Ding projective and the morphism $φ^M$ is a monomorphism. (2) If $T$ is a right coherent ring, $_{B}U$ has finite flat dimension, $U_{A}$ is finitely presented and has finite projective or $FP$-injective dimension, then a right $T$-module $(W_{1}, W_{2})_{φ_{W}}$ is Ding injective if and only if $W_{1}$ and $\ker(\widetilde{φ_{W}})$ are Ding injective and the morphism $\widetilde{φ_{W}}$ is an epimorphism. As a consequence, we describe Ding projective and Ding injective dimensions of a $T$-module.