Certain Rings Whose Simple Singular Modules Are nil-injective

In this paper, we first study some characterizations of left min-abel ring, strongly left min-abel ring and left MC2 ring. Next, we discuss and generalize some well known results for a ring whose simple singular left modules are nil- injective. Finally, as a byproduct of these results we are able to show that if R is a left GMC2 left Goldie ring whose every simple singular left R - module is YJ- injective, then R is a finite product of simple left Goldie ring.

Anahtar Kelimeler:

Left minimal elements, Left min-abel rings, Strongly left min-abel rings, Left MC2 rings, Simple singular modules, Left nil- injective modules

Certain Rings Whose Simple Singular Modules Are nil-injective

Keywords:

Left minimal elements, Left min-abel rings, Strongly left min-abel rings, Left MC2 rings, Simple singular modules, Left nil- injective modules,

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Kim, E. S. Kim, Y. J. and Park, Y. S.: Children ideals and children rings, Kyungpook Math. J. 38, 151-155 (1998).
Michler, G. O. and Villamayor, O. E.: On rings whose simple modules are injective, J. Alg. , 185-201 (1973).
Chen, J. L. and Ding, N. Q.: On generalizations of injectivity, In International Symposium on Ring Theory (Kyongju, 1999),Birkhauser, Boston, Boston, MA, 85-94 (2001).
Wei, J. C.: The rings characterized by minimal left ideal, Acta. Math. Sinica. English. Series. 21:3, 473-482 (2005).
Wei, J. C. and Chen, J. H.: nil- injective rings, in International Electronic J. Algebra, 2, 21 (2007).
Kim, J. Y.: Certain rings whose simple singular modules are GP− injective, Proc. Japan. Acad. 81, 125-128 (2005).
Ding. N. Q. and Chen, J. L.: Rings whose simple singular modulesare Y J− injective, Math Japonica. 40, 191-195 (1994).
Kim, N. K. Nam, S. B. and Kim, J. Y.: On simple singular GP− injective modules, Comm in Alg. 27:5, 2087-2096 (1999).
Roger, Yue Chi Ming.: A note on semi-prime rings, Monatshefte f¨ur Math. 101, 173-182 (1986).
Roger, Yue Chi Ming.: Generalizations of continuous and p− injective modules, Anal. St. Univ. Al.I. CUZA.IASI. 52, 351-364 (2006).
Roger, Yue Chi Ming.: On p− injectivity and generalizations, Riv. Mat. Univ. Parma. 5:5, 188 (1996). Roger, Yue Chi Ming.:
On quasi-Frobeniusean and Artinian rings, Publications De L,institut Math ´ematique. 33, 239-245 (1983).
Roger, Yue Chi Ming.: On semi-prime and reduced rings, Riv. Mat. Univ. Parma. 12, 175 (1986).
Lam, T. Y. and Dugas, A. S.: Quasi-duo rings and stable range descent, J. Pure and Appl. Alg. 195, 243-259 (2005).
Nicholson, W. K. and Sanchez Campos: Rings with the dual of the isomorphism theorem, J. Alg. 271, 391-406. (2004).
Nicholson, W. K. and Watters, J. F.: Rings with projective socle, Proc. Amer. Math. Soc. , 443-450 (1988).
Nicholson, W. K. and Yousif, M. F.: Mininjective ring, J. Alg. 187, 548-578 (1997).
Nicholson. W. K. and Yousif, M. F.: Weakly continuous and C2- rings, Comm in Alg. 29:6, 2446 (2001).
Xue, W. M.: A note on Y J− injectivity, Riv. Mat. Univ. Parma. 6:1, 31-37 (1998).
Lee, Y. and Hun, C.: A Note on Π− regular rings, Kyungpook Math. J. 38, 157-161 (1998). Junchao WEI
School of Mathematics, Yangzhou University, Yangzhou, 225002, P. R. CHINA e-mail jcweiyz@yahoo.com.cn