Popesco-Gabriel theorem for \({\mathcal L}(V)\)
Let R be a ring with identity, \({\mathcal M}_ R\) the category of unital right R-modules, V an injective module in \({\mathcal M}_ R\), and let \({\mathcal L}(V)\) be the full subcategory of \({\mathcal M}_ R\) consisting of all modules of V-dominant dimension \(\geq 2\). The author proves the following ring theoretical version of the Popesco-Gabriel Theorem: Let V be an injective right R-module, \(U\in {\mathcal L}(V)\), \(S=End(U_ R)\), and Q the V-localization of R. Then the following conditions are equivalent: (1) U is a generator for \({\mathcal L}(V)\). (2) V is U-distinguished in \({\mathcal M}_ Q\). (3) The Hom-functors \[ Hom(U_ R,-)_ S: {\mathcal L}(V)\rightleftarrows {\mathcal L}(W): Hom(U^*_ S,-)_ R \] are equivalences, where \(W_ S=Hom(U_ R,V)_ S\) and \({}_ RU^*_ S=_ RHom(U_ R,Q)_ S\). (4) \(Hom(Q/I,V_ R)=0\) with I the trace ideal of U in Q.
scientific article; zbMATH DE number 3950756
scientific article