Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

Query:
Fill in the form and click »Search«...
Format:
Display: entries per page entries
Zbl 0928.16007
Marín, Leandro
Morita equivalence based on contexts for various categories of modules over associative rings.
(English)
[J] J. Pure Appl. Algebra 133, No.1-2, 219-232 (1998). ISSN 0022-4049

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\}.\par 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 $s_1s_2\cdots s_{m_0}\in\text{Im}(\psi)$.\par An example is given of a ring $R$ such that CMOD-$R$ is not equivalent to DMOD-$R$.
[A.I.Kashu (Kishinev)]
MSC 2000:
*16D90 Module categories (assoc. rings and algebras)
18E35 Localization of categories

Keywords: Morita contexts; pairings; category equivalences

Highlights
Master Server

### Zentralblatt MATH Berlin [Germany]

© FIZ Karlsruhe GmbH

Zentralblatt MATH master server is maintained by the Editorial Office in Berlin, Section Mathematics and Computer Science of FIZ Karlsruhe and is updated daily.

Other Mirror Sites

Copyright © 2013 Zentralblatt MATH | European Mathematical Society | FIZ Karlsruhe | Heidelberg Academy of Sciences