Spectrum of a Noncommutative Ring

Abstract

R is any ring with identity. Let Spec r (R) (resp. Spec(R)) be the set of all prime right ideals (resp. all prime ideals) of R and let U r (eR) = {P ∈ Spec r (R) | e ∉ P}. In this article, we study the relationships among various ring-theoretic properties and topological conditions on Spec r (R) (with weak Zariski topology). A ring R is called Abelian if all idempotents in R are central (see Goodearl, 1991). A ring R is called 2-primal if every nilpotent element is in the prime radical of R (see Lam, 2001). It will be shown that for an Abelian ring R there is a bijection between the set of all idempotents in R and the clopen (i.e., closed and open) sets in Spec r (R). And the following results are obtained for any ring R: (1) For any clopen set U in Spec r (R), there is an idempotent e in R such that U = U r (eR). (2) If R is an Abelian ring or a 2-primal ring, then, for any idempotent e in R, U r (eR) is a clopen set in Spec r (R). (3) Spec r (R) is connected if and only if Spec(R) is connected.