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

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 -ring as well as an -ring. In this regard, we prove that R ⋈ I is an -ring if and only if R is an -ring and I is contained in the set of zero divisors Z(R) of R. As to the Property () of R ⋈ I, it turns out that its characterization involves a new concept that we introduce in [6] and that we term the Property () of a module M along an ideal I. In fact, we prove that R ⋈ I is an -ring if and only if R is an -ring, I is an -module along itself and if p is a prime ideal of R such that p ⊆ Z_R(I) ∪ Z¹ (R), then either p ⊆ Z_R(I) or p ⊆ Z¹ (R), where Z¹ (R) := {a ∈ R : a + I ⊆ Z1z(R)}. [ABSTRACT FROM AUTHOR]