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 -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)}.

摘要

本文的主要目的是为了完全表征时被混合的复制ř ⋈我的环的ř沿着理想我是一种形环，以及一个形环。在这方面，我们证明了[R ⋈我是一个形圈当且仅当[R是一个环形和我包含在一套零因子Z（[R的）[R 。对于R ⋈ I 的性质( ) ，事实证明它的表征涉及我们在 [6] 中引入的一个新概念，我们称之为性质 ()沿理想I的模块M。事实上，我们证明R ⋈ I是一个环当且仅当R是一个环，I是一个沿其自身的模并且如果p是R的素理想使得p ⊆ Z _R ( I ) ∪ Z ¹ ( R )，然后p ⊆ Z _R ( I ) 或p ⊆ Z ¹ ( R )，其中 Z ¹ ( R) := {a ∈ R : a + I ⊆ Z1z( R ) }。