Localizations of tensor products

A homomorphism ${\lambda}:A\rightarrow B$ between $R$-modules is called a localization if for all ${\varphi} \in Hom_{R}(A,B)$ there is a unique ${\psi} \in Hom_{R}(B,B)$ such that ${\varphi} ={\psi} \circ {\lambda} $. We investigate localizations of tensor products of torsion-free abelian groups. For example, we show that the natural multiplication map ${\mu}:R\otimes R\rightarrow R$ is a lo cal iza tion if and only if $R$ is an E-ring.