Some aspects on essential Artinian modules and rings

In this paper, we discuss several characterizations of e-Artinian modules in terms of [Formula: see text]-non-summands and proved that if [Formula: see text] is an [Formula: see text]-module satisfying dcc on finitely generated essential non-summands, then every module [Formula: see text] also satisfies the same for any fully invariant submodule [Formula: see text] of [Formula: see text]. Further we prove that for a non-zero [Formula: see text]-local e-Artinian [Formula: see text]-module [Formula: see text] if [Formula: see text] is reduced, then [Formula: see text] is a local module of finite length with respect to essential submodules. Finally we establish the well-known Hilbert Basis Theorem for an e-Artinian ring.