Abstract
The rings considered in this paper are commutative with identity and the modules considered here are modules over commutative rings and are unitary. Let [Formula: see text] be a ring and let [Formula: see text] be a multiplicatively closed subset (m.c. subset) of [Formula: see text]. Let [Formula: see text] be a module over [Formula: see text]. We say that [Formula: see text] is [Formula: see text]-Laskerian (respectively, strongly [Formula: see text]-Laskerian) if [Formula: see text] is an [Formula: see text]-finite [Formula: see text]-module and for any submodule [Formula: see text] of [Formula: see text], either [Formula: see text] or there exist [Formula: see text] (depending on [Formula: see text]) and an [Formula: see text]-decomposable (respectively, a strongly [Formula: see text]-decomposable) submodule [Formula: see text] of [Formula: see text] such that [Formula: see text]. (The concept of an [Formula: see text]-decomposable (respectively, a strongly [Formula: see text]-decomposable) submodule is defined in the introduction.) The aim of this paper is to discuss some basic properties of [Formula: see text]-Laskerian modules.
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