$k$-Congruences and the Zariski topology in semirings

Abstract

The purpose of this paper is to study topological properties of both the set of all $k$-prime ideals and the set of all $k$-prime congruences for any commutative semiring with zero and identity. We first prove that the $k$-prime spectrum, i.e. the set of all $k$-prime ideals equipped with the Zariski topology is a spectral space, and then prove that the set of all $k$-prime congruences is homeomorphic to the $k$-prime spectrum with respect to their Zariski topologies.