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Connection between idempotent radicals of two categories of modules in the situation of adjoint functors. (Russian) Zbl 0604.16004

The aim of this paper is to investigate the relationship between the idempotent radicals of two categories of modules, R-Mod and S-Mod, in the case when these categories are related by a pair (T,H) of adjoint functors. It is shown that the functors T and H define in a canonical way two maps \({\mathcal I}(R)\rightleftarrows^{\alpha '}_{\alpha}{\mathcal I}(S)\) between the class \({\mathcal I}(R)\) of idempotent radicals of R-mod and the class \({\mathcal I}(S)\) of idempotent radicals of S-Mod, and that the maps \(\alpha\) and \(\alpha\) ’ define a Galois connection. Then, the closed objects of this Galois connection are investigated, as well as, when the maps \(\alpha\) and \(\alpha\) ’ are injective. The last part of the paper considers also the case of contravariant adjoint functors between R-mod and S-Mod.
Reviewer: T.Albu

MSC:

16Nxx Radicals and radical properties of associative rings
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
16B50 Category-theoretic methods and results in associative algebras (except as in 16D90)
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