$(n,d)$-COCOHERENT RINGS, $(n,d)$-COSEMIHEREDITARY RINGS AND $(n,d)$-$V$-RINGS

Abstract

Let $R$ be a ring, $n$ be an non-negative integer and $d$ be a positive integer or $\infty$. A right $R$-module $M$ is called \emph{$(n,d)^*$-projective} if ${\rm Ext}^1_R(M, C)=0$ for every $n$-copresented right $R$-module $C$ of injective dimension $\leq d$; a ring $R$ is called \emph{right $(n,d)$-cocoherent} if every $n$-copresented right $R$-module $C$ with $id(C)\leq d$ is $(n+1)$-copresented; a ring $R$ is called \emph{right $(n,d)$-cosemihereditary} if whenever $0\rightarrow C\rightarrow E\rightarrow A\rightarrow 0$ is exact, where $C$ is $n$-copresented with $id(C)\leq d$, $E$ is finitely cogenerated injective, then $A$ is injective; a ring $R$ is called \emph{right $(n,d)$-$V$-ring} if every $n$-copresented right $R$-module $C$ with $id(C)\leq d$ is injective. Some characterizations of $(n,d)^*$-projective modules are given, right $(n,d)$-cocoherent rings, right $(n,d)$-cosemihereditary rings and right $(n,d)$-$V$-rings are characterized by $(n,d)^*$-projective right $R$-modules. $(n,d)^*$-projective dimensions of modules over right $(n,d)$-cocoherent rings are investigated.