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

Authors : Zhu ZHANMIN

Pages : 199-210

Doi:10.24330/ieja.852216

View : 12 | Download : 15

Publication Date : 2021-01-05

Article Type : Research Paper

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{$insert ignore into journalissuearticles values(n,d);^*$-projective} if ${\rm Ext}^1_Rinsert ignore into journalissuearticles values(M, C);=0$ for every $n$-copresented right $R$-module $C$ of injective dimension $\leq d$; a ring $R$ is called \emph{right $insert ignore into journalissuearticles values(n,d);$-cocoherent} if every $n$-copresented right $R$-module $C$ with $idinsert ignore into journalissuearticles values(C);\leq d$ is $insert ignore into journalissuearticles values(n+1);$-copresented; a ring $R$ is called \emph{right $insert ignore into journalissuearticles values(n,d);$-cosemihereditary} if whenever $0\rightarrow C\rightarrow E\rightarrow A\rightarrow 0$ is exact, where $C$ is $n$-copresented with $idinsert ignore into journalissuearticles values(C);\leq d$, $E$ is finitely cogenerated injective, then $A$ is injective; a ring $R$ is called \emph{right $insert ignore into journalissuearticles values(n,d);$-$V$-ring} if every $n$-copresented right $R$-module $C$ with $idinsert ignore into journalissuearticles values(C);\leq d$ is injective. Some characterizations of $insert ignore into journalissuearticles values(n,d);^*$-projective modules are given, right $insert ignore into journalissuearticles values(n,d);$-cocoherent rings, right $insert ignore into journalissuearticles values(n,d);$-cosemihereditary rings and right $insert ignore into journalissuearticles values(n,d);$-$V$-rings are characterized by $insert ignore into journalissuearticles values(n,d);^*$-projective right $R$-modules. $insert ignore into journalissuearticles values(n,d);^*$-projective dimensions of modules over right $insert ignore into journalissuearticles values(n,d);$-cocoherent rings are investigated.
Keywords : cocoherent ring, cosemihereditary ring, V ring