Rings for which every cosingular module is projective

Yıl 2019, Cilt: 48 Sayı: 4, 973 - 984, 08.08.2019

Y. Talebi , A. R. M. Hamzekolaee , M. Hosseinpour A. Harmanci , B. Ungor

Öz

Let $R$ be a ring and $M$ be an $R$-module. In this paper we investigate modules $M$ such that every (simple) cosingular $R$-module is $M$-projective. We prove that every simple cosingular module is $M$-projective if and only if for $N\leq T\leq M$, whenever $T/N$ is simple cosingular, then $N$ is a direct summand of $T$. We show that every simple cosingular right $R$-module is projective if and only if $R$ is a right $GV$-ring. It is also shown that for a right perfect ring $R$, every cosingular right $R$-module is projective if and only if $R$ is a right $GV$-ring. In addition, we prove that if every $\delta$-cosingular right $R$-module is semisimple, then $\overline{Z}(M)$ is a direct summand of $M$ for every right $R$-module $M$ if and only if $\overline{Z}_{\delta}(M)$ is a direct summand of $M$ for every right $R$-module $M$.

Anahtar Kelimeler

projective module, cosingular module, $\delta$-cosingular module, $GV$-ring

Kaynakça

• [1] A.N. Abyzov, Weakly regular modules, Russian Math. 48 (3), 1-3, 2004.
• [2] J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting Modules. Supplements and Projectivity in Module Theory, Frontiers in Mathematics, Boston, Birkh¨auser, 2006.
• [3] N.V. Ding, D.V. Huynh, P.F. Smith and R. Wisbauer, Extending Modules, Pitman Research Notes in Mathematics Series 313 Harlow: Longman Scientific, 1996.
• [4] F. Kasch and A. Mader, Rings, Modules and the Totals, Frontiers in Mathematics, Birkh¨auser, 2004.
• [5] D. Keskin, N. Orhan, P. Smith and R. Tribak, Some rings for which the cosingular submodule of every module is a direct summand, Turk. J. Math. 38, 649-657, 2014.
• [6] D. Keskin and R. Tribak, When M-cosingular modules are projective, Vietnam J. Math. 33 (2), 214-221, 2005.
• [7] G.O. Michler and O.E. Villamayor, On rings whose simple modules are injective, J. Algebra 25, 185-201, 1973.
• [8] S.H. Mohamed and B.J.Müller, Continuous and Discrete Modules, London Math. Soc. Lecture Notes Series 147, Cambridge, University Press, 1990.
• [9] A.C. Özcan, On $GCO$-modules and $M$-small modules, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 51 (2), 25-36, 2002.
• [10] A.C. Özcan, The torsion theory cogenerated by $\delta$-$M$-small modules and $GCO$-modules, Comm. Algebra 35 (2), 623-633, 2007.
• [11] V.S. Ramamurthy and K.M. Rangaswamy, Generalized $V$-rings, Math. Scand. 31, 69-77, 1972.
• [12] Y. Talebi and M.J. Nematollahi, Modules with $C^{*}$-condition, Taiwanese J. Math. 13 (5), 1451-1456, 2009.
• [13] Y. Talebi and N. Vanaja, The torsion theory cogenerated by $M$-small modules, Comm. Algebra 30 (3), 1449-1460, 2002.
• [14] R. Tribak and D. Keskin, On $\overline{Z}_M$-semiperfect modules, East-West J. Math. 8 (2), 193-203, 2006.
• [15] R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Reading, 1991.