Cyclic covering of a module over an Artinian ring

Abstract

Given a commutative ring with identity [Formula: see text] and an [Formula: see text]-module [Formula: see text], a subset [Formula: see text] of [Formula: see text] is a cyclic covering of [Formula: see text], if this module is the union of the cyclic submodules [Formula: see text], where [Formula: see text]. Such covering is said to be irredundant, if no proper subset of [Formula: see text] is a cyclic covering of [Formula: see text]. In this work, an irredundant cyclic covering of [Formula: see text] is constructed for every Artinian commutative ring [Formula: see text]. As a consequence, a cyclic covering of minimal cardinality of [Formula: see text] is obtained for every finite commutative ring [Formula: see text], extending previous results in the literature.