On Modules for Which All Submodules Are Projection Invariant and the Lifting Condition

Abstract

The notion of projection invariant subgroups was first introduced by Fuchs in [7]. We will define the module- theoretic version of the projection invariant subgroup. Let R be a ring and M a right R - module. We call a submodule N of M the projection invariant if every projection pi of M onto a direct summand maps N into itself, i.e. N is invariant under any projection of M. In this note, we give several characterizations to these class of modules that generalize the recent results in [14]. We also define and study the PI- lifting modules which is a generalization of FI-lifting module. It is shown that if each M i is a PI-lifting module for all 1 <= i <= n, then M = circle plus(n)(i=1) M-i is a PI-lifting module. In particular, we focus on rings satisfying the following condition: (*) Every submodule of M is projection invariant. We prove that if R has the (*) property, then R circle plus R does not satisfy the (*) property.