Lifting Modules with Indecomposable Decompositions

Abstract

A module M is called a “lifting module” if, any submodule A of M contains a direct summand B of M such that A/B is small in M/B. This is a generalization of projective modules over perfect rings as well as the dual of extending modules. It is well known that an extending module with ascending chain condition (a.c.c.) on the annihilators of its elements is a direct sum of indecomposable modules. If and when a lifting module has such a decomposition is not known in general. In this article, among other results, we prove that a lifting module M is a direct sum of indecomposable modules if (i) rad(M ^(I)) is small in M ^(I) for every index set I, or, (ii) M has a.c.c. on the annihilators of (certain) elements, and rad(M) is small in M.

Keywords:

• Decomposition
• Extending module
• Indecomposable module
• Lifting module
• Projective module

Mathematics Subject Classification:

• Primary 16D10, 16D70, 16D80
• Secondary 16D40, 16P60

ACKNOWLEDGMENTS

This article was prepared during the second author's visit to the Center of Ring Theory and Its Applications at Ohio University. She gratefully acknowledges the support of the center and the support from TUBITAK (Turkish Scientific Research Council). Both authors wish to express their thanks to Professors D. V. Huynh and S. K. Jain of Ohio University for their kind help during the preparation of this article. We also thank the referee for suggestions improving the article.

Notes

Communicated by M. Ferrero.