Applications of Duality to the Pure-injective Envelope

Abstract

Given an R-T-bimodule R K T and an R-S-bimodule R M S , we study how properties of R K T affect the K-double dual M * * = Hom T [Hom R (M, K ), K], considered as a right S-module. If R K is a cogenerator, then for every R-S-bimodule, the natural morphism M : M → M * * is a pure-monomorphism of right S-modules. If R K is the minimal (injective) cogenerator and K T is quasi-injective, then M * * is a pure-injective right S-module. If R K is the minimal (injective) cogenerator, and T = End R K, it is shown that K T is quasi-injective if and only if the K-topology on R is linearly compact. If the R K-topology on R is of finite type, then the natural morphism R : R → R * * is the pure-injective envelope of R R as a right module over itself.Key words bicommutator · M-topology · minimal injective cogenerator · pure-injective envelope · quasi-injective Mathematics Subject Classifications (2000) 16D50 · 16D90 · 16S90Let R and T be associative rings with identity, and R K T an R-T-bimodule. The bimodule R K T induces an additive endofunctor (−) * * : R-Mod → R-Mod, the Kdouble dual, of the category of left R-modules. It is defined byThere is a natural transformation : 1 R−Mod → (−) * * from the identity functor to the K-double dual, given by the evaluation morphism M : M → M * * , M (m) : ξ → ξ(m). The functorial nature of (−) * * implies that if M has an R-S-bimodule