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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: $\text{CMOD-}R=\{M_R\mid M\simeq\Hom_R(R,M)$ 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. $\forall(r_n)_{n\in\bbfN}\in R^\bbfN\ \exists n_0\in\bbfN$ such that $r_1r_2\cdots r_{n_0}\in\text{Im}(\varphi)$, $\forall(s_m)_{m\in\bbfN}\in S^\bbfN\ \exists m_0\in\bbfN$ 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-$R$.

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