Topological properties of the prime spectrum of a semimodule

Abstract

The objective of this paper is to study topological properties of the prime spectrum of a unitary semimodule over a semiring with zero and nonzero identity. We define a top semimodule over a semiring, show that the prime spectrum is a T0-space, and prove that each irreducible closed subset of the prime spectrum has a generic point and that the prime spectrum is a compact space if the top semimodule is finitely generated. For a multiplication semimodule over a commutative semiring, we reveal the structure of the radical of a subsemimodule and prove that the multiplication semimodule is finitely generated iff the prime spectrum is a compact space, that in the prime spectrum, every basic open set is compact and the intersection of finitely many basic open sets is also compact, and that the prime spectrum is a spectral space if the multiplication semimodule is finitely generated.