Properties of the subtractive prime spectrum of a semimodule

Abstract

For a top semimodule over a semiring with zero and nonzero identity, this paper studies the interplay between topological properties of the subtractive prime spectrum and algebraic properties of the semimodule. We prove that the subtractive prime spectrum of the subtractively finitely generated top semimodule is a compact space, and establish necessary and sufficient conditions for the top semimodule to be subtractively finitely generated. For a multiplication semimodule over a commutative semiring, we prove that the radical of a subtractive subsemimodule coincides with its subtractive radical, that every proper subtractive subsemimodule is contained in a subtractive prime subsemimodule, that the multiplication semimodule is subtractively finitely generated iff its subtractive prime spectrum is a compact space, that in the subtractive prime spectrum, the intersection of finitely many basic open sets is compact, and that the subtractive prime spectrum of the subtractively finitely generated multiplication semimodule is a spectral space.