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

Year 2021, , 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

References

• F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, 2nd ed., Graduate Texts in Mathematics, 13, Springer-Verlag, New York, 1992.
• D. Bennis, H. Bouzraa and A.-Q. Kaed, On $n$-copresented modules and $n$-co-coherent rings, Int. Electron. J. Algebra, 12 (2012), 162-174.
• D. L. Costa, Parameterizing families of non-noetherian rings, Comm. Algebra, 22(10) (1994), 3997-4011.
• V. A. Hiremath, Cofinitely generated and cofinitely related modules, Acta Math. Acad. Sci. Hungar., 39(1-3) (1982), 1-9.
• J. P. Jans, On co-noetherian rings, J. London Math. Soc., 1(2) (1969), 588-590.
• R. W. Miller and D. R. Turnidge, Factors of cofinitely generated injective modules, Comm. Algebra, 4(3) (1976), 233-243.
• R. Wisbauer, Foundations of Module and Ring Theory, Algebra, Logic and Applications, 3, Gordon and Breach Science Publishers, Philadelphia, PA, 1991.
• W. M. Xue, On co-semihereditary rings, Sci. China Ser. A., 40(7) (1997), 673-679.
• W. M. Xue, On n-presented modules and almost excellent extensions, Comm. Algebra, 27(3) (1999), 1091-1102.
• Z. M. Zhu, On n-coherent rings, n-hereditary rings and n-regular rings, Bull. Iranian Math. Soc., 37(4) (2011), 251-267.
• Z. M. Zhu, n-cocoherent rings, n-cosemihereditary rings and n-V -rings, Bull. Iranian Math. Soc., 40(4) (2014), 809-822.
• Z. M. Zhu and J. L. Chen, FCP-projective modules and some rings, J. Zhejiang Univ. Sci. Ed., 37(2) (2010), 126-130.