Category of $n$-FCP-gr-projective modules with respect to special copresented graded modules

Year 2022, , 157 - 175, 16.07.2022

Mostafa Amını Driss Bennıs Soumia Mamdouhı

https://doi.org/10.24330/ieja.1077596

Cited By: 1

Abstract

Let $R$ be a ring graded by a group $G$ and $n\geq1$ an integer. We introduce the
notion of $n$-FCP-gr-projective $R$-modules and by using of
special finitely copresented graded modules, we investigate that (1) there exist some equivalent characterizations of $n$-FCP-gr-projective modules and graded right modules of $n$-FCP-gr-projective dimension
at most $k$ over $n$-gr-cocoherent rings, (2) $R$ is right $n$-gr-cocoherent if and only if for every short exact sequence $0 \rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ of graded right $R$-modules, where $B$ and $C$ are $n$-FCP-gr-projective, it follows that $A$ is $n$-FCP-gr-projective if and only if ($gr$-$\mathcal{FCP}_{n}$, $gr$-$\mathcal{FCP}_{n}^{\bot}$) is a hereditary cotorsion theory, where $gr$-$\mathcal{FCP}_n$ denotes the class
of $n$-FCP-gr-projective right modules. Then we examine some of the conditions equivalent to that each right $R$-module is $n$-FCP-gr-projective.

Keywords

$n$-gr-Cocoherent ring, special gr-copresented module, $n$-FCP-gr-projective module

References

• M. Amini, Gorenstein $\pi[T]$-projectivity with respect to a tilting module, Int. Electron. J. Algebra, 27 (2020), 114-126.
• M. Amini and F. Hassani, Copresented dimension of modules, Iran. J. Math. Sci. Inform., 14(2) (2019), 139-151.
• M. J. Asensio, J. A. López Ramos and B.Torrecillas, Covers and envelopes over gr-Gorenstein rings, J. Algebra, 215 (1999), 437-459.
• M. J. Asensio, J. A. López Ramos and B.Torrecillas, FP-gr-injective modules and gr-FC-ring, Algebra and number theory (Fez), 1–11, Lecture Notes in Pure and Appl. Math., 208, Dekker, New York, 2000.
• D. Bennis, H. Bouzraa and A. Kaed, On $n$-copresented modules and $n$-cocoherent rings, Int. Electron. J. Algebra, 12 (2012), 162-174.
• J. Chen and N. Ding, On $n$-coherent rings, Comm. Algebra, 24(10) (1996), 3211-3216.
• D. L. Costa, Parameterizing families of non-Noetherian rings, Comm. Algebra, 22(10) (1994), 3997-4011.
• D. E. Dobbs, S. Kabbaj and N. Mahdou, $n$-coherent rings and modules, Lecture Notes in Pure and Appl. Math., 185 (1997), 269-281.
• J. R. Garcia Rozas, J. A. Lopez- Ramos and B. Torrecillas, On the existence of flat covers in $R$-gr, Comm. Algebra, 29(8) (2001), 3341-3349.
• R. Hazrat, Graded Rings and Graded Grothendieck Groups, London Mathematical Society Lecture Note Series, 2016.
• J. P. Jans, On co-Noetherian rings, J. London Math. Soc., (2)1 (1969), 588-590.
• M. Kleiner and I. Reiten, Abelian categories, almost split sequences, and comodules, Trans. Amer. Math. Soc., 357(8) (2005), 3201-3214.
• L. Mao, Ding-graded modules and gorenstein $gr$-flat modules, Glasg. Math. J., 60(2) (2018), 339-360.
• C. Nastasescu, Some constructions over graded rings, J. Algebra, 120(1) (1989), 119-138.
• C. Nastasescu and F.Van Oystaeyen, Graded Ring Theory, North-Holland Mathematical Library 28, North-Holland Publishing Company, Amsterdam, 1982.
• C. Nastasescu and F.Van Oystaeyen, Methods of Graded Rings, Lecture Notes in Mathematics, 1836, Springer-Verlag, Berlin, 2004.
• J. Rotman, An Introduction to Homological Algebra, Second edition, Universitext, Springer, New York, 2009.
• X. Yang and Z. Liu, FP-gr-injective modules, Math. J. Okayama Univ., 53 (2011), 83-100.
• J. Zhan and Z. Tan, Finitely copresented and cogenerated dimensions, Indian J. Pure Appl. Math., 35(6) (2004), 771-781.
• T. Zhao, Z. Gao and Z. Huang, Relative FP-gr-injective and gr-flat modules, Internat. J. Algebra Comput., 28(6) (2018), 959-977.
• Z. Zhu, $n$-cocoherent rings, $n$-cosemihereditary rings and $n$-$V$-rings, Bull. Iranian Math. Soc., 40(4) (2014), 809-822.
• R. Wisbauer, Foundations of Module and Ring Theory, Revised and translated from the 1988 German edition. Algebra, Logic and Applications, 3, Gordon and Breach Science Publishers, Philadelphia, PA, 1991.
• W. Xue, On almost excellent extensions, Algebra Colloq., 3(2) (1996), 125-134.
• W. Xue, On $n$-presented modules and almost excellent extensions, Comm. Algebra, 27(3) (1999), 1091-1102.