2-primal weak (σ,δ)-rigid rings

Abstract

For a ring R, an endomorphism σ of R and δ a σ-derivation of R, we introduce a weak (σ,δ)-rigid ring, which generalizes the notion of (σ,δ)-rigid rings and investigate its properties. Moreover, we state and prove a necessary and sufficient condition for a weak (σ,δ)-rigid ring to be a (σ,δ)-rigid ring. We prove that a (σ,δ)-ring is a weak (σ,δ)-rigid ring and conversely that the prime radical of a weak(σ,δ)-rigid ring is a (σ,δ)-ring. We also find a relation between minimal prime ideals and completely prime ideals of a ring R, where R is a (σ,δ)-ring and R is a 2-primal weak (σ,δ)-rigid ring.