On trivial extensions which are quasi-Frobenius ones
In this short note it is proved that the trivial extension \(R\ltimes M\) of a ring R by a bimodule \(_ RM_ R\) is a left quasi-Frobenius extension of R in the sense of \textit{B. Müller} [Math. Z. 85, 345-368 (1964; Zbl 0203.044)] if and only if there exists an idempotent e in the center C of R such that the eC-module \(M^ R=\{m\in M| mr=rm\) for all \(r\in R\}\) is finitely generated and projective of rank 1, and \(M\cong M^ R\otimes_ CR\). This result generalizes a similar characterization by \textit{Y. Kitamura} [Arch. Math. 34, 111-113 (1980; Zbl 0508.16013)] for \(R\ltimes M\) to be a Frobenius extension of R.
scientific article; zbMATH DE number 3842073