On Quasi-Rigid Ideals and Rings

Abstract

Abstract. Let σ be an endomorphism and I a σ-ideal of a ring R. Pearson and Stephenson called I a σ-semiprime ideal if whenever A is an ideal of R and m is an integer such that Aσ t (A) ⊆ I for all t ≥ m, then A ⊆ I, where σ is an automorphism, and Hong et al. called I a σ-rigid ideal if aσ(a) ∈ I implies a ∈ I for a ∈ R. Notice that R is called a σ-semiprime ring (resp., a σ-rigid ring) if the zero ideal of R is a σ-semiprime ideal (resp., a σ-rigid ideal). Every σ-rigid ideal is a σ-semiprime ideal for an automorphism σ, but the converse does not hold, in general. We, in this paper, introduce the quasi σ-rigidness of ideals and rings for an automorphism σ which is in between the σ-rigidness and the σ-semiprimeness, and study their related properties. A number of connections between the quasi σ-rigidness of a ring R and one of the Ore extension R[x; σ, δ] of R are also investigated. In particular, R is a (principally) quasi-Baer ring if and only if R[x; σ, δ] is a (principally) quasi-Baer ring, when R is a quasi σ-rigid ring. DefinitionsLet σ be an endomorphism of a ring R, the additive map δ :