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Baer and Morita duality. (English) Zbl 0974.16004

The notion of a Baer duality is defined as a triple \((R,{_RU_T},T)\) consisting of rings \(R,T\) and a bimodule \(_RU_T\) faithful on both sides such that \({\mathcal L}(_RR)\) and \({\mathcal L}(U_T)\), as well as \({\mathcal L}(_RU)\) and \({\mathcal L}(T_T)\), are anti-isomorphic, where \({\mathcal L}(X)\) denotes the lattice of submodules of a module \(X\). Although no specific assumptions are required on the nature of anti-isomorphisms in a Baer duality, it is shown that annihilation induces the anti-isomorphism between \({\mathcal L}(_RR)\) and \({\mathcal L}(U_T)\), as well as \({\mathcal L}(_RU)\) and \({\mathcal L}(T_T)\).
The first three sections of the paper are devoted to find conditions which ensure the existence of anti-isomorphisms between the lattices of submodules of two fixed modules. Grothendieck’s condition AB5* is also used to characterize Baer dualities. An alternative characterization of anti-isomorphisms between \({\mathcal L}(_RM)\) and \({\mathcal L}(N_T)\) is given in terms of “injectivity conditions”. Some examples of Baer dualities are given.
In the fourth section the authors apply their results to linearly compact modules and Morita duality, and show a new link between linear compactness and injectivity when a lattice anti-isomorphism is given. Finally, in the fifth section, the authors introduce a concept of paired idempotents. The AB5* condition for these pairs is investigated.
Reviewer: Y.Kurata (Hadano)

MSC:

16D90 Module categories in associative algebras
16D50 Injective modules, self-injective associative rings
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