Polynomial extensions of quasi-Baer rings

Abstract

For a ring endomorphism α and an α-derivation δ, we introduce α-compatible rings which are a generalization of α-rigid rings, and study on the relationship between the quasi Baerness and p.q.-Baer property of a ring R and those of the polynomial extensions (including formal skew power series, skew Laurent polynomials and skew Laurent series). As a consequence we obtain a generalization of [6], [8] and [16].

On Commutativity of Skew Polynomials at Zero

“…There exists another generalization of α-rigid rings. For an endomorphism α of a ring R, the ring R is called an α-compatible ring [7] if for each a, b ∈ R, ab = 0 ⇔ aα(b) = 0. In this case, clearly the endomorphism α is a monomorphism.…”

On Annihilations of Ideals in Skew Monoid Rings

“…According to Krempa [34], an endomorphism α of a ring R is said to be rigid if aα(a) = 0 implies a = 0 for a ∈ R. A ring R is said to be α-rigid if there exists a rigid endomorphism α of R. In [21], the authors introduced α-compatible rings and studied their properties. A ring R is α-compatible if for each a, b ∈ R, ab = 0 if and only if aα(b) = 0.…”

Semicommutative Property on Nilpotent Products

“…According to Krempa [16], an endomorphism σ of a ring R is called rigid if aσ(a) = 0 implies a = 0 for a ∈ R. Hong et al [10] called R a σ-rigid ring if there exists a rigid endomorphism σ of R. Note that any rigid endomorphism of a ring is a monomorphism and σ-rigid rings are reduced rings by [10,Proposition 5]. Following [9], a ring R is called σ-compatible if for each a, b ∈ R, ab = 0 ⇔ aσ(b) = 0, and R is called δ-compatible if for each a, b ∈ R, ab = 0 ⇒ aδ(b) = 0. If R is both σ-compatible and δ-compatible, then R is called (σ, δ)-compatible, in this case the endomorphism σ is clearly a monomorphism.…”