Idempotent functors and localizations in categories of modules and Abelian groups

Abstract

The present paper contains various results on idempotent functors and localizations in categories of modules and Abelian groups. CONTENTS

Colocalizations and dualities for modules and Abelian groups

“…The group Hom S (R, A) is an R-module and a pullback S-module. We take the value mapping v : Hom S (R, A) → A, v(α) = α(1), α ∈ Hom S (R, A), which is an S-homomorphism (see the text before Proposition 1.5 in [14]). We consider methods to obtain some colocalization of the module A with the use of v. We also denote by v the restriction of the homomorphism v to the above defined T(e)-radical W Hom S (R, A) of the R-module Hom S (R, A).…”

“…(2) If R is a T-ring, then every R-module is a T-module [14,Theorem 2.6]. Consequently, W H(R) = H(R) and we can use (1).…”

“…In the category of torsion-free Abelian groups, a duality is considered.This paper is close to [14], where we consider idempotent coaugmentation functors in categories of modules and localizations of modules. By dualizing these notions, we obtain idempotent augmentation functors and colocalizations; they are also called cellular covers in literature.…”

“…By dualizing these notions, we obtain idempotent augmentation functors and colocalizations; they are also called cellular covers in literature. Similarly to [14], it is possible and desirable to study general properties of idempotent augmentation functors and colocalizations; we can also study the problem of preserving by them ring or module structures, and so on. Here we only consider some standard colocalizations (see Secs.…”

“…The factor module R/e(S) is denoted by R 0 . All R-modules are assumed to be pullback S-modules.We present several definitions from [14]. An R-module A is called a T(e)-module or a T-module with respect to the homomorphism e if the mapping A ⊗ S R → A ⊗ R R, a ⊗ S r → a ⊗ R r, is an isomorphism.…”