On projection-invariant submodules of QTAG-modules

A module M over an associative ring R with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules. Here we study projection-invariant submodule of QTAG-module. A submodule N of a QTAG-module M is said to be projection-invariant in M if f(N)⊆N, for all idempotent endomorphisms f in End(M). Clearly, fully invariant submodules are projection-invariant. Mehdi et. al. characterized fully invariant submodules and characteristic submodules with the help of their socles. Here we investigate the socles of projection-invariant submodules of QTAG-modules.