Galois Connections Between Lattices of Preradicals Induced by Adjoint Pairs Between Categories of Modules

Abstract

We extend the result which establishes that every equivalence between categories of modules induces an isomorphism between the corresponding lattices of preradicals, proving that every adjoint pair between categories of modules induces a Galois connection between the corresponding lattices of preradicals. We prove some general properties of this Galois connection. In particular, given an ideal I of a ring R we study the Galois connection induced by the natural projection R→R/I and we prove that it yields a partition of R-pr into intervals, which are described. We study this partition in the case of the ring \(\mathbb {Z}\) and ideals \(p^{n}\mathbb {Z}\) and use it to locate the idempotent radicals on abelian groups.