On ADS Modules and Rings

Abstract

A right module M over a ring R is said to be ADS if for every decomposition M = S ⊕ T and every complement T′ of S, we have M = S ⊕ T′. In this article, we study and provide several new characterizations of this new class of modules. We prove that M is semisimple if and only if every module in σ[M] is ADS. SC and SI rings also characterized by the ADS notion. A ring R is right SC-ring if and only if every 2-generated singular R-module is ADS.

Key Words:

• ADS module
• ADS ring
• CS module
• Fully-invariant module

2010 Mathematics Subject Classification:

• 16D50
• 16D60
• 16D80

ACKNOWLEDGMENTS

The first author was supported by National Foundation for Science and Technology Development of Vietnam. The authors would like to thank the referee for the very helpful comments and suggestions.

Notes

Communicated by E. Puczylowski.