Gelfand factor rings and weak Zariski topologies

摘要

Let R be any ring with identity. Let N (R) ( resp. J (R)) denote the prime radical (resp. Jacobson radical) of R, and let Spec(r) (R) ( resp. Spec(l) (R), Max(r) (R), Prim(r) (R)) denote the set of all right prime ideals ( resp. all left prime ideals, all maximal right ideals, all right primitive ideals) of R. In this article, we study the relationships among various ring-theoretic properties and topological conditions on Spec(r) (R) ( with weak Zariski topology). The following results are obtained: ( 1) R/N(R) is a Gelfand ring if and only if Spec(r) (R) is a normal space if and only if Spec(l) (R) is a normal space; (2) R/J(R) is a Gelfand ring if and only if every right prime ideal containing J(R) is contained in a unique maximal right ideal.