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

Year 2021, Volume: 29 Issue: 29, 199 - 210, 05.01.2021

Zhu Zhanmın

https://doi.org/10.24330/ieja.852216

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.

Keywords

$(n,d)$-cocoherent ring, $(n,d)$-cosemihereditary ring, $(n,d)$-$V$-ring, $(n,d)^*$-projective module

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