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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anderson F W, Fuller K R. Rings and Categories of Modules. Grad Texts in Math, Vol 13. New York: Springer-Verlag, 1992
Chatters A W, Hajarnavis C R. Rings in which every complement right ideal is a direct summand. Q J Math, 1977, 28: 61–80
Chen J L, Li W X. On artiness of right CF rings. Comm Algebra, 2004, 32(11): 4485–4494
Clark J, Lomp C, Vanaja N, Wisbauer R. Lifting Modules: Supplements and Projectivity in Module Theory. Front Math. Basel: Birkhäuser-Verlag, 2006
Dung N V, Huynh D V, Smith P F, Wisbauer R. Extending Modules. Pitman Research Notes Math Ser, 313. Essex: Longman Scientific & Technical, 1994
Faith C, Huynh D V. When self-injective rings are QF: a report on a problem. J Algebra Appl, 2002, 1(1): 75–105
Gómez Pardo J L, Guil Asensio P A. Essential embedding of cyclic modules in projectives. Trans Amer Math Soc, 1997, 349(11): 4343–4353
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
Goodearl K R. Ring Theory: Nonsingular Rings and Modules. Monogr Pure Appl Math, Vol 33. New York: Dekker, 1976
Lam T Y. Lectures on Modules and Rings. Grad Texts in Math, Vol 189. New York: Springer-Verlag, 1998
Lam T Y. A First Course in Noncommutative Rings. Grad Texts in Math, Vol 131. New York: Springer-Verlag, 2001
Mohamed S H, Müller B J. Continuous and Discrete Modules. London Math Soc Lecture Note Ser, Vol 147. Cambridge: Cambridge Univ Press, 1990
Nicholson W K, Yousif M F. Quasi-Frobenius Rings. Cambridge Tracts in Math, Vol 158. Cambridge: Cambridge Univ Press, 2003
Shen L. An approach to the Faith-Menal conjecture. Int Electron J Algebra, 2007, 1(1): 46–50
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Shen, L., Li, W. Generalization of CS condition. Front. Math. China 12, 199–208 (2017). https://doi.org/10.1007/s11464-016-0596-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11464-016-0596-x