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

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 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 Mi is a PI-lifting module for all 1 ? i ? n, then M = ?n i=1Mi 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 ? R does not satisfy the (*) property

On (weakly) co-Hopfian automorphism-invariant modules

© 2020, © 2020 Taylor & Francis Group, LLC. A module M over a ring R is called automorphism-invariant if M is invariant under automorphisms of its injective hull E(M) and M is called co-Hopfian if each injective endomorphism of M is an automorphism. It is shown that (1) being co-Hopfian, directly-finite and having the cancelation property or the the substitution property are all equivalent conditions on automorphism-invariant modules, (2) if (Formula presented.) is an automorphism-invariant module, then M is co-Hopfian iff M1 and M2 are co-Hopfian, (3) if M is an automorphism-invariant module, then M is co-Hopfian if and only if E(M) is co-Hopfian. The module M is called weakly co-Hopfian if any injective endomorphism of M is essential. We also show that (4) if R is a right and left automorphism-invariant ring, then R is stably finite iff RR or (Formula presented.) is a co-Hopfian module iff if Rn is weakly co-Hopfian as a right or left R-module for all (Formula presented.)