Morita equivalence based on contexts for various categories of modules over associative rings (Q1295636)

For an associative ring \(R\) (possibly without identity) various subcategories of the category of all (right) \(R\)-modules MOD-\(R\) are considered, in particular: \(\text{CMOD-}R=\{M_R\mid M\simeq\Hom_R(R,M)\) canonically\}, \(\text{DMOD-}R=\{M_R\mid M\otimes_RR\simeq M\) canonically\}. Every Morita context between \(R\) and \(S\) with epimorphic pairings induces the equivalences \(\text{CMOD-}R\simeq \text{CMOD-}S\) and \(\text{DMOD-}R\simeq\text{DMOD-}S\). The converse of this fact is proved under hypotheses weaker than the surjectivity of pairings. Namely, for every Morita context \((R,S,P,Q,\varphi,\psi)\) 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) \(P\otimes_R-\) and \(Q\otimes_S-\) are inverse category equivalences between the categories \(R\)-DMOD and \(S\)-DMOD; (3) the given context is left acceptable, i.e. \(\forall(r_n)_{n\in\mathbb{N}}\in R^\mathbb{N}\;\exists n_0\in\mathbb{N}\) such that \(r_1r_2\cdots r_{n_0}\in\text{Im}(\varphi)\), \(\forall(s_m)_{m\in\mathbb{N}}\in S^\mathbb{N}\;\exists m_0\in\mathbb{N}\) such that \(s_1s_2\cdots s_{m_0}\in\text{Im}(\psi)\). An example is given of a ring \(R\) such that CMOD-\(R\) is not equivalent to DMOD-\(R\).