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Costar modules. (English) Zbl 0990.16009

The authors introduce a new type of modules which induce generalizations of Morita duality and are, in some sense, dual to the \(*\)-modules of C. Menini and A. Orsatti [Rend. Semin. Mat. Univ. Padova 82, 203-231 (1989; Zbl 0701.16007)], whence the name ‘costar modules’. A module \(Q_R\) with endomorphism ring \(S=\text{End}(Q_R)\) is a costar module whenever the \(_SQ_R\)-duals induce a duality between the class of torsionless right \(R\)-modules whose \(Q\)-duals are finitely generated over \(S\) and the class of finitely generated torsionless left \(S\)-modules. It is shown that the class of costar modules contains other classes of modules that induce generalizations of Morita duality, namely, quasi-duality modules, and cotilting modules, and also that, over a finite dimensional algebra \(k\), finitely generated costar modules are just the \(k\)-duals of \(*\)-modules. In the last part of the paper, a natural strengthening of costar modules, called strongly costar modules, is considered. The resulting class not only contains all cotilting modules, but also all the bimodules that induce generalized Morita dualities in the sense of R. R. Colby [Commun. Algebra 17, No. 7, 1709-1722 (1989; Zbl 0677.16026)].

MSC:

16D90 Module categories in associative algebras
16D20 Bimodules in associative algebras
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