Self injective amalgamated duplication of a ring along an ideal (Q2853969)

Let \( I\) be an ideal of a ring \(R\). Let \(R\bowtie I\) be the amalgamated duplication of the ring \(R\) along the ideal \(I\), i.e., \(R\bowtie I := \{(r, r + i) \mid r \in R \text{ and } i \in I\} \;(\subseteq R \times R)\). The main result of the present paper can be stated as follows: \(R\bowtie I\) is a self injective ring (i.e., a ring that is an injective module over itself) if and only if \(R\) is a self injective ring and there exists an idempotent element \(e\in R\) such that \(I = eR\).