Self Injective Amalgamated Duplication of a Ring Along an Ideal

Abstract

In this paper, we give a characterization for the amalgamated duplication of a ring R along an ideal I, denoted by R ⋈ I, to be self-injective.

Prüfer Conditions in an Amalgamated Duplication of a Ring Along an Ideal

“…By the fact that every ideal over a Von Neumann regular ring is pure, we conclude from Theorem 2.1 the following Corollary which have already proved in [3] with different methods. A simple example of Theorem 2.1 is given by introducing the notion of the trace of modules.…”

“…On the other hand, Maimani and Yassemi, in [16], have studied the diameter and girth of the zero-divisor of the ring R ⊲⊳ I. Recently in [3], the authors study some homological properties of the rings R ⊲⊳ I. Some references are [4,5,6,16].…”

A Note on Amalgamated Rings Along an Ideal

“…The exception is [12], where associative unital rings are considered. For more details see for example [2,3,6,9,11,13].…”

Self injective property in amalgamated algebra along an ideal

“…Self-injective rings (i.e., rings that are injective modules over themselves) play an important role in ring theory since they have connections with several kinds of rings; e.g., quasi-Frobenius rings, semiprimary rings, and Kasch rings (see [12]). In [5], The authors characterize an amalgamated duplication of a ring R along an ideal I, denoted by R ⊲⊳ I to be self-injective.…”

“…In this paper, we investigate the transfer of self-injective and quasi-Frobenius properties to amalgamation A ⊲⊳ f J and so we generalize [5].…”