SS-supplemented modules

Abstract

A module M is called ss-supplemented if every submodule U of M has a

supplement V in M such that U(intersection)  V is semisimple. It is shown that a finitely generated

module M is ss-supplemented if and only if it is supplemented and Rad(M) (submodule) Soc(M). A module

M is called strongly local if it is local and Rad(M) (submodule)  Soc(M). Any direct sum of strongly

local modules is ss-supplemented and coatomic. A ring R is semiperfect and Rad(R) (submodule)

Soc(_RR) if and only if  every left R-module is ss-supplemented if and only if _RR is a finite sum of strongly local

submodules.

Keywords

• strongly local module,semisimple module,ss-supplemented module

References

1. Alizade, R., Bilhan, G. and Smith, P.F., Modules whose maximal submodules have supplements, Communications in Algebra, 29(6) (2001) 2389-2405.
2. Büyükaşık, E., Mermut, E. and Özdemir, S., Rad-supplemented modules, Rend. Sem. Mat. Univ. Padova 124 (2010) 157-177.
3. Kasch, F., Modules and Rings, London New York, 1982.
4. Lomp, C., On semilocal modules and rings, Communications in Algebra 27(4) (1999) 1921-1935.
5. Mohamed, S.H., Müller, B.J., Continuous and Discrete Modules, London Math. Soc. LNS 147 Cambridge University, 1990.
6. Sharpe, D.W., Vamos, P., Injective Modules, Cambridge University Press, Cambridge, 1972.
7. Wisbauer, R., Foundations of Module and Ring Theory, Gordon and Breach, 1991
8. Zhou, D. X., Zhang, X.R., Small-Essential Submodules and Morita Duality, Southeast Asian Bulletin of Mathematics 35 (2011) 1051-1062.

9. Zöschinger, H., Moduln die in jeder Erweiterung ein Komplement haben, Mathematica Scandinavica 35 (1974) 267-287.
10. Zöschinger, H., Komplementierte moduln über Dedekindringen, Journal of Algebra 29 (1974) 42-56.