Generalization of CS condition

doi:10.1007/s11464-016-0596-x

Front. Math. China

2017, Vol. 12

Issue (1) : 199-208 https://doi.org/10.1007/s11464-016-0596-x

RESEARCH ARTICLE

Generalization of CS condition

Liang SHEN¹(

),Wenxi LI²

¹. Department of Mathematics, Southeast University, Nanjing 210096, China
². Department of Mathematics and Physics, Anhui University of Technology, Ma’anshan 243002, China

Download: PDF(147 KB)
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

Abstract

Let R be an associative ring with identity. An R-module M is called an NCS module if C(M)∩S(M)={0}, where C(M) and S(M) denote the set of all closed submodules and the set of all small submodules of M, respectively. It is clear that the NCS condition is a generalization of the well-known CS condition. Properties of the NCS conditions of modules and rings are explored in this article. In the end, it is proved that a ring R is right Σ-CS if and only if R is right perfect and right countably Σ-NCS. Recall that a ring R is called right Σ-CS if every direct sum of copies of R_R is a CS module. And a ring R is called right countably Σ-NCS if every direct sum of countable copies of R_R is an NCS module.

Keywords NCS modules NCS rings CS rings Σ-CS rings countably Σ-NCS

Corresponding Author(s): Liang SHEN

Issue Date: 17 November 2016

Cite this article:

Liang SHEN,Wenxi LI. Generalization of CS condition[J]. Front. Math. China, 2017, 12(1): 199-208.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-016-0596-x
https://academic.hep.com.cn/fmc/EN/Y2017/V12/I1/199

1	Anderson F W, Fuller K R. Rings and Categories of Modules. Grad Texts in Math, Vol 13. New York: Springer-Verlag, 1992 https://doi.org/10.1007/978-1-4612-4418-9
2	Chatters A W, Hajarnavis C R. Rings in which every complement right ideal is a direct summand. Q J Math, 1977, 28: 61–80 https://doi.org/10.1093/qmath/28.1.61
3	Chen J L, Li W X. On artiness of right CF rings. Comm Algebra, 2004, 32(11): 4485–4494 https://doi.org/10.1081/AGB-200034189
4	Clark J, Lomp C, Vanaja N, Wisbauer R. Lifting Modules: Supplements and Projectivity in Module Theory. Front Math. Basel: Birkhäuser-Verlag, 2006
5	Dung N V, Huynh D V, Smith P F, Wisbauer R. Extending Modules. Pitman Research Notes Math Ser, 313. Essex: Longman Scientific & Technical, 1994
6	Faith C, Huynh D V. When self-injective rings are QF: a report on a problem. J Algebra Appl, 2002, 1(1): 75–105 https://doi.org/10.1142/S0219498802000070
7	Gómez Pardo J L, Guil Asensio P A. Essential embedding of cyclic modules in projectives. Trans Amer Math Soc, 1997, 349(11): 4343–4353 https://doi.org/10.1090/S0002-9947-97-01529-8
8	Gómez Pardo J L, Guil Asensio P A. Torsionless modules and rings with finite essential socle. In: Dikranjan D, Salce L, eds. Abelian Groups, Module Theory, and Topology. Lect Notes Pure Appl Math, Vol 201. New York: Marcel Dekker, 1998, 261–278
9	Goodearl K R. Ring Theory: Nonsingular Rings and Modules. Monogr Pure Appl Math, Vol 33. New York: Dekker, 1976
10	Lam T Y. Lectures on Modules and Rings. Grad Texts in Math, Vol 189. New York: Springer-Verlag, 1998
11	Lam T Y. A First Course in Noncommutative Rings. Grad Texts in Math, Vol 131. New York: Springer-Verlag, 2001 https://doi.org/10.1007/978-1-4419-8616-0
12	Mohamed S H, Müller B J. Continuous and Discrete Modules. London Math Soc Lecture Note Ser, Vol 147. Cambridge: Cambridge Univ Press, 1990 https://doi.org/10.1017/CBO9780511600692
13	Nicholson W K, Yousif M F. Quasi-Frobenius Rings. Cambridge Tracts in Math, Vol 158. Cambridge: Cambridge Univ Press, 2003 https://doi.org/10.1017/CBO9780511546525
14	Shen L. An approach to the Faith-Menal conjecture. Int Electron J Algebra, 2007, 1(1): 46–50

Viewed

Full text

Abstract

Cited

Shared

Discussed