On regularity of rings (Q2762285)

Throughout, \(R\) is an associative ring with identity and modules are unitary. A right \(R\)-module \(M\) is GP-injective if, for any \(0\neq a\in R\), there exists a positive integer \(n\) with \(a^n\neq 0\) such that any \(R\)-homomorphism from \(a^nR\) to \(M\) extends to one from \(R\) to \(M\). The authors prove that \(R\) is (von Neumann) regular if and only if every cyclic left \(R\)-module is GP-injective, if and only if \(R\) is left PP and left GP-injective. Yue Chi Ming raised the question: Is a left PP right P-injective ring regular? The authors answer this in the negative. This example appeared in the reviewer's paper [in Riv. Mat. Univ. Parma (6) 1, 31-37 (1998; Zbl 0929.16002)], and was essentially given by \textit{J. Clark} [in C. R. Math. Acad. Sci., Soc. R. Can. 18, No. 2-3, 85-87 (1996; Zbl 0865.16008)].