Cotilting versus pure-injective modules

Let R and S be arbitrary associative rings. A left R-module RW is said to be cotilting if the class of modules cogenerated by RW coincides with the class of modules for which the func- tor Ext1R(−, W ) vanishes. In this paper we characterize the cotilting modules which are pure-injective. The two notions seem to be strictly connected: Indeed all the examples of cotilting modules known in the literature are pure-injective. We observe that if RWS is a pure-injective cotilting bimodule, both R and S are semiregular rings and we give a characteri- zation of the reflexive modules in terms of a suitable “linear compactness” notion.