\textit{RD}-injectivity of tensor products of modules (Q2833636)

Let \(R\) be a commutative associative ring with unity. An exact sequence of \(R\)-modules \(0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0\) is \textbf{RD-pure} if it remains exact when tensoring it with an \(R\)-module of the form \(R/Rr\) for each \(r\in R\). An \(R\)-module \(E\) is \textbf{RD-injective} if it is injective relatively to any RD-pure sequence of \(R\)-modules. The following question is studied in this paper: let \(M\) and \(N\) be RD-injective \(R\)-modules, is \(M\otimes_RN\) RD-injective too? When \(R\) is a ring for which the answer to this question is positive and \(R^{(\Lambda)}\) an RD-injective \(R\)-module for each set \(\Lambda\) it is proven that each direct sum of RD-injective modules is RD-injective. In this case \(R\) is the product of a pure-semisimple ring with a finite quasi-Frobenius ring. When each simple module is RD-injective, it is shown that the tensor product of any two Artinian modules is RD-injective.NEWLINENEWLINEIt is possible to improve Theorem 2.10. Since \(R \) is a semiprime Goldie ring its quotient ring \(Q \) is semisimple. So, each torsion-free p-injective \(R \)-module \(M \) is a \(Q \)-module. Consequently, for each \(R \)-module \(N \), \(M\otimes_RN \) is a \(Q \)-module, whence it is injective.NEWLINENEWLINEReviewer's remark: Proposition 2.15 is correct but not its proof, because [\textit{A. Moradzadeh-Dehkordi}, J. Algebra 460, 128--142 (2016; Zbl 1357.16013), Theorem 2.1] is false. But \(M\) and \(N \) are direct summands of finite direct sums of cyclic modules. Hence \(M\otimes_RN \) is also a direct summand of a finite direct sum of cyclic modules.