Lifting Modules Over Right Perfect Rings

A module M is called extending if, for any submodule X of M, there exists a direct summand of M which contains X as an essential submodule, that is, for any submodule X of M, there exists a closure of X in M which is a direct summand of M. Dually, a module M is said to be lifting if, for any submodule X of M, there exists a direct summand of M which is a co-essential submodule of X, that is, for any submodule X of M, there exists a co-closure of X in M which is a direct summand of M. Okado (1984) has studied the decomposition of extending modules over right noetherian rings. He obtained the following: A ring R is right noetherian if and only if every extending R-module can be expressed as a direct sum of indecomposable (uniform) modules. In this article, we show that every (finitely generated) lifting module over a right perfect (semiperfect) ring can be expressed as a direct sum of indecomposable modules. And we consider some application of this result.

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