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 T 0 -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.

中文翻译：

---

半模素谱的拓扑性质

摘要 本文的目的是研究具有零和非零身份的半环上酉半模素谱的拓扑性质。我们在半环上定义一个顶半模，证明素谱是一个 T 0 空间，并证明素谱的每个不可约闭子集都有一个泛点，如果顶半模是有限生成。对于交换半环上的乘法半模，我们揭示了子半模的根的结构，并证明乘法半模是有限生成的，当当素谱是一个紧空间，在素谱中，每个基本开集都是紧的，并且有限多个基本开集的交集也是紧的，