On property (A) of the amalgamated duplication of a ring along an ideal

Abstract

The main purpose of this paper is to totally characterize when the amalgamated duplication R ⋈ I of a ring R along an ideal I is an A-ring as well as an SA-ring. In this regard, we prove that R ◃▹ I is an SA-ring if and only if R is an SA-ring and I is contained in the set of zero divisors Z(R) of R. As to the Property (A) of R ⋈ I, it turns out that its characterization involves a new concept that we introduce in [6] and that we term the Property (A) of a module M along an ideal I. In fact, we prove that R ⋈ I is an A-ring if and only if R is an A-ring, I is an A-module along itself and if p is a prime ideal of R such that p ⊆ Z_R(I)∪Z^I(R), then either p ⊆ Z_R(I) or p ⊆ Z^I (R), where Z^I (R) := {a ∈ R : a + I ⊆ Z(R)}.