A note on a Cohen-type theorem for w-Artinian modules

In this note, we prove that a [Formula: see text]-module [Formula: see text] is [Formula: see text]-Artinian if and only if it is [Formula: see text]-cofinitely generated and for every prime [Formula: see text]-ideal [Formula: see text] of [Formula: see text] with [Formula: see text], there exists a [Formula: see text]-submodule [Formula: see text] of [Formula: see text] such that [Formula: see text] is [Formula: see text]-cofinitely generated and [Formula: see text], where [Formula: see text] Besides, we show that every principal ideal is not a [Formula: see text]-ideal for some Noetherian rings.