Localizations of tensor products. (Q396503)

Let \(A\) and \(B\) be modules. A homomorphism \(\lambda\colon A\to B\) is called a localization if for all \(\varphi\in\Hom(A,B)\) there is a unique \(\psi\in\Hom(B,B)\) such that \(\varphi=\psi\circ\lambda\). The authors investigate localizations of tensor products of torsion-free Abelian groups. For example, they show that if \(R\) is a torsion-free ring, the natural multiplication \(\mu\colon R\otimes R\to R\) is a localization if and only if \(R\) is an \(E\)-ring. Their results are applied to present some surprising properties of tensor powers of torsion-free Abelian groups, for example strongly indecomposable groups \(G\) for which \(G\otimes G\) is reduced and completely decomposable.