On Coreflexive Coalgebras and Comodules over Commutative Rings

In this note we study dual coalgebras of algebras over arbitrary (noetherian) commutative rings. We present and study a generalized notion of coreflexive comodules and use the results obtained for them to characterize the so called coreflexive coalgebras. Our approach in this note is an algebraically topological one. Introduction The concept of coreflexive coalgebras was studied, in the case of commutative base fields, by several authors. An algebraic approach was presented by E. Taft ([Taf72], [Taf77]), while a topological one was presented mainly by D. Radford ([HR74], [Rad73]) and studied by several authors (e.g. [Miy75], [Wit79]). In this note we present and study a generalized concept of coreflexive comodules and use it to characterize coreflexive coalgebras over commutative (noetherian) rings. In particular we generalize results from the papers mentioned above from the case of base fields to the case of arbitrary (noetherian) commutative ground rings. Throughout this paper R denotes a commutative ring with 1R 6= 0R. We consider R as a left and a right linear topological ring with the discrete topology. The category of R-(bi)modules will be denoted by MR. The unadorned −⊗− and Hom mean −⊗R− and HomR respectively. For an R-module M, an R-submodule K ⊂ M will be called N-pure for some R-module N, if the canonical R-linear mapping ιK ⊗ idN : K⊗R N → M ⊗R N is injective. We call K ⊂ M pure (in the sense of Cohn), if it’s N -pure for every R-module N. For every R-module L, we denote with L the algebraic dual R-module of all R-linear maps from L to R. ∗MSC (2000): 16D90, 16W30, 16Exx