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Morita equivalence based on contexts for various categories of modules over associative rings. (English) Zbl 0928.16007

For an associative ring R (possibly without identity) various subcategories of the category of all (right) R-modules MOD-R are considered, in particular: CMOD-R={M R MHom R (R,M) canonically}, DMOD-R={M R M R RM canonically}.

Every Morita context between R and S with epimorphic pairings induces the equivalences CMOD-RCMOD-S and DMOD-RDMOD-S. The converse of this fact is proved under hypotheses weaker than the surjectivity of pairings. Namely, for every Morita context (R,S,P,Q,φ,ψ) the following conditions are equivalent: (1) Hom R (P,-) and Hom S (Q,-) are inverse category equivalences between the categories CMOD-R and CMOD-S; (2) P R - and Q S - are inverse category equivalences between the categories R-DMOD and S-DMOD; (3) the given context is left acceptable, i.e. (r n ) n R n 0 such that r 1 r 2 r n 0 Im(φ), (s m ) m S m 0 such that s 1 s 2 s m 0 Im(ψ).

An example is given of a ring R such that CMOD-R is not equivalent to DMOD-R.

MSC:
16D90Module categories (associative rings and algebras); Morita equivalence and duality
18E35Localization of categories