Stabilization of Kurosh chains
Let K be a commutative ring with identity, and \({\mathcal A}^ a \)class of K-algebras closed under subobjects and quotient objects. Let \({\mathcal M}\) be a subclass of \({\mathcal A}\), and \(\tilde {\mathcal M}\) the smallest radical subclass of \({\mathcal A}\) containing \({\mathcal M}\). This class can be constructed as the union of the so called Kurosh chain (\({\mathcal M}_{\alpha})\) (indexed by ordinals \(\alpha\geq 0)\) for \({\mathcal M}\). One says that the Kurosh chain (\({\mathcal M}_{\alpha})_{\alpha \geq 0}\) of \({\mathcal M}\) stabilizes at the place \(\alpha\) if \(\alpha\) is the smallest ordinal such that \({\mathcal M}_{\alpha}={\mathcal M}_{\alpha +1}\), and then \(\tilde {\mathcal M}={\mathcal M}_{\alpha}\). The aim of this paper is to prove the following result: If the class \({\mathcal A}\) contains all commutative K- algebras and \(\alpha\) is an ordinal such that \(\alpha\leq \omega\), where \(\omega\) is the first transfinite ordinal, then there exists a subclass \({\mathcal M}\) of \({\mathcal A}\) such that the Kurosh chain of \({\mathcal M}\) stabilizes at the place \(\alpha\). There is a mistake in the definition of the first class \({\mathcal M}_ 0\) in the Kurosh chain (\({\mathcal M}_{\alpha})_{\alpha \geq 0}:\) \({\mathcal M}_ 0\) is the class of all epimorphic images of algebras in \({\mathcal M}\) (and not in \({\mathcal A}\), as is stated in the paper).
scientific article; zbMATH DE number 3906666