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

Year 2021, , 199 - 210, 05.01.2021
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.

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.
Year 2021, , 199 - 210, 05.01.2021
https://doi.org/10.24330/ieja.852216

Abstract

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.
There are 12 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Zhu Zhanmın This is me

Publication Date January 5, 2021
Published in Issue Year 2021

Cite

APA Zhanmın, Z. (2021). $(n,d)$-COCOHERENT RINGS, $(n,d)$-COSEMIHEREDITARY RINGS AND $(n,d)$-$V$-RINGS. International Electronic Journal of Algebra, 29(29), 199-210. https://doi.org/10.24330/ieja.852216
AMA Zhanmın Z. $(n,d)$-COCOHERENT RINGS, $(n,d)$-COSEMIHEREDITARY RINGS AND $(n,d)$-$V$-RINGS. IEJA. January 2021;29(29):199-210. doi:10.24330/ieja.852216
Chicago Zhanmın, Zhu. “$(n,d)$-COCOHERENT RINGS, $(n,d)$-COSEMIHEREDITARY RINGS AND $(n,d)$-$V$-RINGS”. International Electronic Journal of Algebra 29, no. 29 (January 2021): 199-210. https://doi.org/10.24330/ieja.852216.
EndNote Zhanmın Z (January 1, 2021) $(n,d)$-COCOHERENT RINGS, $(n,d)$-COSEMIHEREDITARY RINGS AND $(n,d)$-$V$-RINGS. International Electronic Journal of Algebra 29 29 199–210.
IEEE Z. Zhanmın, “$(n,d)$-COCOHERENT RINGS, $(n,d)$-COSEMIHEREDITARY RINGS AND $(n,d)$-$V$-RINGS”, IEJA, vol. 29, no. 29, pp. 199–210, 2021, doi: 10.24330/ieja.852216.
ISNAD Zhanmın, Zhu. “$(n,d)$-COCOHERENT RINGS, $(n,d)$-COSEMIHEREDITARY RINGS AND $(n,d)$-$V$-RINGS”. International Electronic Journal of Algebra 29/29 (January 2021), 199-210. https://doi.org/10.24330/ieja.852216.
JAMA Zhanmın Z. $(n,d)$-COCOHERENT RINGS, $(n,d)$-COSEMIHEREDITARY RINGS AND $(n,d)$-$V$-RINGS. IEJA. 2021;29:199–210.
MLA Zhanmın, Zhu. “$(n,d)$-COCOHERENT RINGS, $(n,d)$-COSEMIHEREDITARY RINGS AND $(n,d)$-$V$-RINGS”. International Electronic Journal of Algebra, vol. 29, no. 29, 2021, pp. 199-10, doi:10.24330/ieja.852216.
Vancouver Zhanmın Z. $(n,d)$-COCOHERENT RINGS, $(n,d)$-COSEMIHEREDITARY RINGS AND $(n,d)$-$V$-RINGS. IEJA. 2021;29(29):199-210.