Document Type: Research Paper

Author

Department of Mathematics, South Tehran Branch, Islamic Azad University, Tehran, Iran

Abstract

Let $R$ be a ring, $\sigma$ be an endomorphism of $R$ and $M_R$ be a $\sigma$-rigid module. A module $M_R$ is called quasi-Baer if the right annihilator of a principal submodule of $R$ is generated by an idempotent. It is shown that an $R$-module $M_R$ is a quasi-Baer module if and only if $M[[x]]$ is a quasi-Baer module over the skew power series ring $R[[x,\sigma]]$.

Keywords

Main Subjects

[1] D. D. Anderon, V. Camilo, Armendariz rings and gaussian rings. Comm. Algebra 26 (1998) 2265-2275.

[2] S. Aninn, Associated primes over skew polynomials rings. Comm. Algebra 30 (2002) 2511-2528.

[3] E. P. Armendariz, A note on extensions of Baer and p.p.rings. J. Austral. Math. Soc. 18 (1974) 470-473.

[4] G. F. Birkenmeier, J. Y. Kim, J. K. Park, On quasi-Baer rings. Algebra and it applications (Athens, OH) (1999) 67-92.

[5] G. F. Birkenmeier, J. Y. Kim, J. K. Park, Principally quasi-Baer rings. Comm. Algebra 29 (2011) 639-660.

[6] G. F. Birkenmeier, J. Y. Kim, J. K. Park, Polynomial extensions of Baer and quasi-Baer rings. J. Pure Appl. Algebra 159 (2001) 25-42.

[7] G. F. Birkenmeier, Idempotents and completely semiprime ideals. Comm. Algebra. 11 (1983) 567-580.

[8] J. W. Brewer, Power series over commutative rings. Lecture Notes in Pure and Applied Mathematics, 64, Marcel Dekker, Inc., New York 1981.

[9] A. W. Chatters, C. R. Hajarnavis, Rings with chain conditions. Research Nots in Mathematics. 44, Pitman, Boston, Mas.-London, 1980.

[10] W. E. Clark, Twited matrix units semigroup algebras. Duke Math. J. 34 (1967) 417-424

[11] E. Hashemi, A. Moussavi, Polynomial extensions of quasi-Baer rings. Acta Math. Hungar. 107 (2005) 207-224.

[12] Y. Hirano, On annihilator ideals of a polynomial ring over a non commutative ring. J. Pure Appl. Algebra. 168 (2002) 45-52.

[13] C. Y. Hong. N. K. Kim, T. K. Kwak, On skew Armendariz rings. Comm. Algebra. 31 (2003) 103-122.

[14] C. Huh, Y. Lee, Smoktunowicz, Agata, Armendariz rings and semicommutativ rings. Comm. Algebra. 30 (2002) 751-761.

[15] I. Kaplansky, Rings of operators. W. A. Benjamin, Inc., New York, 1968.

[16] N. K. Kim, K. H. Lee, Y. Lee, Power eris rings satisfying a zero divisor property. Comm. Algebra. 34 (2006) 2205-2218.

[17] N. K. Kim, Y. Lee, Armendariz rings and reduced rings. J. Algebra 223 (2000) 447-488.

[18] J. Krempa, Some examples of reduced rings. Algebra Colloq. 3 (1996) 289-300.

[19] T. K. Lee, Y. Zhou, Armendariz and reduced rings. Comm. Algebra. 32 (2004) 2287-2299.

[20] T. K. Lee, Y. Zhou, Reduced modules. Rings, modules, algebras and abelian groups. 365-377. Lecture Notes in Pure and Appl. Math., 236. Marcel Dekker, New York, 2004.

[21] Z. Liu, A note on principally quai-Baer rings. Comm. Algebra. 30 (2002) 3885-3890.

[22] M. B. Rege, S. Chhawchharia, Armendariz rings. Proc. Japan Acad. Ser. A Math. Sci. 73 (1997) 14-17.

[23] L. Zhongkui, Z. Renyu, A generalization of p.p.-rings and p.q.-Baer rings. Glasgow Math. J. 48 (2006) 217-229.