A generalization of lifting modules

Abstract

In this paper we introduce the notions of G 1L-module and G 2L-module which are two proper generalizations of -lifting modules. We give some character- izations and properties of these modules. We show that a G 2 L-module decomposes into a semisimple submodule M1 and a submodule M2 of M such that every non-zero submodule of M2 contains a non-zero -cosingular submodule. Throughout this article, all rings are associative with an identity, and all modules are unitary right R modules. Let M be an R-module. By NM(N � � M) we mean that N is a submodule (direct summand) of M. A submodule N of a module M is called essential in M if for every nonzero submodule L of M, N L 0 (denoted by Ne M) and A submodule N of a module M is called small in M if for every proper submodule L of M, N +L M (denoted by N ≪ M). A module M is called hollow if every proper submodule of M is small in M. M is called a small module if there exists a module T such that M ≪ T. If N=K ≪ M=K, then K is called a cosmall submodule of N in M. A submodule N of M is called coclosed if N has no proper cosmall submodule. Recall that the singular submodule Z(M) of a module M is the set of m 2 M with mI = 0 for some essential ideal I of R. If Z(M) = M (Z(M) = 0), then M is called a singular (non-singular) module. Let K, N be submodules of M. Following (14), as a