On localizations in Morita contexts

A Morita context $(R,{}_RU_S,{}_SV_R,S)$ with a mapping $(\;{,}\;)\colon U\otimes_SV\to R$ defines for every $M\in{}_R\mathscr M$ a canonical homomorphism $\varphi_M\colon M\to \operatorname{Hom}_S(V,\operatorname{Hom}_R(U,M))$. Necessary and sufficient conditions are found for $\varphi_M$ to be an $r_I$-localization of the module $M$ for every $M\in{}_R\mathscr M$, where $r_I$ is the ideal torsion defined by the ideal $I=(U,V)$ of the ring $R$. In particular, these conditions are satisfied when ${}_R(U\otimes _SV)$ is a projective module with trace $I$.
Bibliography: 9 titles.