Radicals of rings and pullbacks
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.
scientific article; zbMATH DE number 719440
scientific article