Ideals And Topological Properties Of Semirings

In this work our definition of Semiring is given without zero 0 and unity 1. The focus of the study is to give appropriate evaluation on Semirings which have their own related properties with Distributive lattices. We prove that for Arithmetic and Regular Semirings with unity 1 prime and irreducible ideal are equivalent. For arbitrary Semiring with unity 1 we have been able to give that every ideal to be intersection of irreducible ideals containing it. We work on commutative Hemirings for understanding Connectedness and Compactness of spectrum in terms of Zariski topology and the smallest Alexandroff topologzz containing zz. We give Connectedness and Compactness characterization of Topology zz on Gelfand Semiring relating to be Semi local and Local Semirings. And for Semirings in general we give the equivalency between Irreducibility of and connectedness of zz.rnKeywords: Hemiring, Alexandroff topology, Zariski topology, idempotent Semiring, Gelfand Semiring, m-Semiring, Arithmetic Semiring, Regular Semiring, prime spectrum

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