Abstract
Given an R-T-bimodule R K T and 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 Φ 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 Φ R : R → R** is the pure-injective envelope of R R as a right module over itself.
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The author is partially supported by NSF Grant DMS-02-00698.
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Herzog, I. Applications of Duality to the Pure-injective Envelope. Algebr Represent Theor 10, 135–155 (2007). https://doi.org/10.1007/s10468-006-9039-9
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DOI: https://doi.org/10.1007/s10468-006-9039-9