Mostafa AMINI, Driss BENNIS, Soumia MAMDOUHI

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

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,

PDF

___

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.

International Electronic Journal of Algebra-Cover

ISSN: 1306-6048
Yayın Aralığı: Yılda 2 Sayı
Başlangıç: 2007
Yayıncı: Abdullah HARMANCI

Arşiv