Localizations and adjointness
The results are the logical continuation of the author's paper [Mat. Issled. 65, 71-87 (1982; Zbl 0492.16006)]. The homomorphisms \(\Phi_ M:M\to Hom_ S(U,U\otimes_ RM)\) and \(\Psi_ N:U\otimes_ RHom_ S(U,N)\to N\) for \({}_ SU_ R\) some S-R-module, \(M\in_ R{\mathcal M}\) the category of left R-modules and \(N\in_ S{\mathcal M}\) the category of left S-modules are under consideration. The author shows, that the homomorphism \(\Phi_ M\) is an \(r_ 0\)- localization of M \((r_ 0\) is the radical defined by \(r_ 0(M)=\ker \Phi_ M)\) if and only if \(\Psi_{U\otimes_ RM}\) is an isomorphism. The homomorphism \(\Psi_ N\) is an \(s_ 0\)-colocalization of a module N \((s_ 0\) is an idempotent preradical, for which \(s_ 0(N)=im \Psi_ N)\) if and only if \(\Phi_{Hom_ S(U,N)}\) is an isomorphism.
scientific article; zbMATH DE number 3851301