Generators of existence varieties of regular rings and complemented Arguesian lattices
Let \({\mathcal A}_r\) denote the class of simple Artinian (von Neumann) regular unital rings. Any existence variety \(\mathcal V\) of regular rings is generated by \({\mathcal V}\cap{\mathcal A}_r\) (cf. [\textit{C. Herrmann} and \textit{M. Semenova}, J. Algebra 314, No. 1, 235--251 (2007; Zbl 1139.06003)]). The main result (5.4) of the paper characterizes classes \({\mathcal C} \subset {\mathcal A}_r\) of the form (*) \({\mathcal V}\cap{\mathcal A}_r\) by a set of five conditions concerning the division rings belonging to \(\mathcal C\). In more detail, for a given positive integer \(n\) and a division ring \(D\) let \(R(D,n)\) denote the ring of \(n\times n\)-matrices over \(D\). Furthermore, given a class \({\mathcal C}\) contained in \({\mathcal A}_r\) they define the class \(D_n({\mathcal C})\) of those division rings \(D\) which fulfil \(R(D,n)\in {\mathcal C}\). \(\mathcal C\) is called \textit{closed} iff the following five conditions hold: (i) \(D_n({\mathcal C})\) is closed under ultraproducts for \(n < \omega\); (ii) \(D_n({\mathcal C})\subseteq D_m({\mathcal C})\) for \(m\leq n < \omega\); (iii) if \(n=mk\) and \(F\in D_n({\mathcal C})\) and \(D\) is a division subring of \(R(F,n)\) then \(D\in D_m({\mathcal C})\); (iv) if \(D_n({\mathcal C})\) always contains a division ring of prime characteristic \(p\), then any division ring of characteristic \(p\) is contained in \(\bigcap\{D_n({\mathcal C}) : 0<n<\omega\}\); (v) \(D_1({\mathcal C})\) is the class of all division rings. Now, closedness of \(\mathcal C\) characterizes the situation (*). An analogous notion of closedness is used in (4.4) to characterize classes \({\mathcal C} \subset {\mathcal A}_l\) of the form \({\mathcal V}\cap{\mathcal A}_l\), where \({\mathcal A}_l\) denotes the class of simple Arguesian sectionally complemented lattices of finite length, and \(\mathcal V\) an existence variety of Arguesian sectionally complemented lattices.
scientific article; zbMATH DE number 5900618
scientific article