Baer duality for commutative rings (Q2769469)

Let \(R\) and \(T\) be rings, and let \(_RU_T\) be a bimodule faithful on both sides. The triple \((R,{_RU_T},T)\) is a Baer duality if \({\mathcal L}({_RR})\) and \({\mathcal L}(U_T)\), as well as \({\mathcal L}({_RU})\) and \({\mathcal L}(T_T)\), are respectively anti-isomorphic, where \({\mathcal L}(X)\) denotes the submodule lattice of a module \(X\). If, in addition, \(R\) and \(T\) are isomorphic rings, then \(R\) is Baer self-dual. Morita (self-)dualities are natural examples of Baer (self-)dualities and other examples are dual rings.NEWLINENEWLINENEWLINEThe authors apply the general theory of Baer duality developed previously to commutative rings. It is shown that if \(R\) is a commutative ring having a Baer duality, then \(R\) is Baer self-dual. In the second section criteria to decide whether a commutative AB5* ring has Baer self-duality are given and it is shown that any commutative AB5* domain has Baer self-duality. In the final section, the authors explore the relation between AB5* and linear compactness, and the relation between Baer duality and Morita duality.