A characterization of the nil radical of a ring

Abstract

It is shown that all primitive ideals are of the form p = P(Γ, S, ψ) and that a ring R is nil if and only if it has no prime ideals of the form P = P(Γ9 S, ψ). It is shown that the nil radical of any ring is the intersection of ail prime ideals P = P(Γ, S, φ). It is shown that if P = P(Γ, S, φ) for all prime ideals P Q R then the nil and Baer radicals coincide for all homomorphic images of R. If the nil and Baer radicals coincide for all homomorphic images of R, it is shown that any prime ideal P of R is contained in a prime ideal P' — Pr(Γ, S, φ). Finally, by consideration of prime ideals P = P(Γ, S, φ), two theorems are proved giving information about rings satisfying very special conditions.