On co-semihereditary rings (Q1373843)

Semihereditary rings can be defined by requiring either (i) finitely generated submodules of finitely generated projective modules to be projective or (ii) finitely generated submodules of (arbitrary) projective modules to be projective. The equivalence of (i) and (ii) follows since every finitely generated submodule of a projective module can be realized as a submodule of a finitely generated projective module. However the equivalence of the duals of (i) and (ii) was left as an open problem by \textit{R. W. Miller} and \textit{D. R. Turnidge} [in Commun Algebra 4, 233-243 (1976; Zbl 0322.16011)]\ and they asked if every finitely cogenerated factor module of an arbitrary injective module can be realized as a factor of a finitely cogenerated injective. The duals of (i) and (ii) produce the following definitions. A ring \(R\) is called left co-semihereditary if every finitely cogenerated factor of a finitely cogenerated injective left \(R\)-module is injective. The ring \(R\) is called strongly left co-semihereditary if every finitely cogenerated factor of an arbitrary injective left \(R\)-module is injective. In response to these questions, the paper under review begins with an example of a finitely cogenerated factor of an injective module which is not a factor of any finitely cogenerated injective and an example of a left co-semihereditary ring which is not strongly left co-semihereditary. In the second section the author shows that if a bimodule \(_RU_S\) defines a Morita duality then \(R\) is left co-semihereditary if and only if \(S\) is right semihereditary. In the third section he shows that if the ring \(S\) is a finite normalizing extension of \(R\) with \(S_R\) projective, \(_RS\) flat and \(S\) right \(R\)-projective, then \(S\) is (strongly) right co-semihereditary if and only if \(R\) is. The final section gives an internal characterization of (strongly) left co-semihereditary rings.