The prime submodules hypergraph of a free module of finite rank over a commutative ring

Abstract

Let [Formula: see text] be a commutative ring with identity, [Formula: see text] be a prime ideal of [Formula: see text] and [Formula: see text] be a free [Formula: see text]-module of finite rank. In this paper, we use hypergraphs to give a full characterization of prime submodules of [Formula: see text]. Also we define a hypergraph [Formula: see text] called the prime submodules hypergraph of [Formula: see text] with respect to [Formula: see text] and use properties of Steiner systems for counting the number of [Formula: see text]-prime submodules of [Formula: see text] when the number of cosets of [Formula: see text] in [Formula: see text] is finite.