Radicals of rings and pullbacks
type: Item
instance of: scholarly article
title: Radicals of rings and pullbacks
zbMATH Open document ID: 0821.16021
DOI: 10.1016/0022-4049(93)00005-P
author: George Janelidze
author: L. Márki
published in: Journal of Pure and Applied Algebra
publication date: 1995-08-10T00:00:00Z
review text: The paper provides a purely category theoretical framework for radical theory, done by characterizing semisimple classes in terms of pull backs. Let \(\mathcal X\) be a class of associative rings for which the inclusion functor into the category of all rings has a left adjoint \(I\) (equivalently, \(\mathcal X\) is subdirectly closed). Denoting by \(\eta\) the unit of the adjunction, if for every ring \(A\) and surjective \(f : A \to X \in {\mathcal X}\) there exists a unique \(\overline{f} : IA \to X\) with commutative diagram \[ \begin{tikzcd} A \ar[rr,"\eta^A"]\ar[dr, "f"'] && IA \ar[dl,"\overline{f}"] \\{} & X & {}\end{tikzcd} \] then \(\overline{f}\) is said to be well defined. If \(\alpha : A \to B\) is such that \(I\alpha = \eta_ B \alpha : IA \to IB\) is well defined, then \(I\alpha\) is said to be well defined. The class \(\mathcal X\) is then a semisimple class if and only if \[ \begin{tikzcd} IA \ar[r,"\overline{f}"] \ar[d,"I\alpha"'] & X \ar[d,"\varphi"]\\ IB \ar[r,"\overline{g}"'] & Y\end{tikzcd} \] is a pullback with well defined \(I\alpha\). Also three more characterizations are given.
Mathematics Subject Classification ID: 16N80
Mathematics Subject Classification ID: 18E40
zbMATH DE Number: 719440
zbMATH Keywords: radicals
zbMATH Keywords: semisimple classes
zbMATH Keywords: pullbacks
zbMATH Keywords: left adjoints
zbMATH Keywords: subdirectly closed classes
reviewed by: R. Wiegandt
MaRDI profile type: Publication
cites work: Q3992448
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description: scientific article; zbMATH DE number 719440