On Steinitz properties
The authors are interested in discussing various `Steinitz properties' of rings \(R\) with identity, their interrelationships and connections with `other' classes of rings described in different manners. Steinitz properties of rings \(R\) with identity are variants of the following pattern (modules are unital): \(R\) is \((A)(B)\) Steinitz if any \((C)\) subset of a \((D)\) free \((A)\) \(R\)-module \(F\) can be extended to a basis by adjoining elements of an \((E)\) subset. Thus, in the original definition of left Steinitz rings one has: \((A)\equiv\) left; \((B)\) unused; \((C)\) linearly independent; \((D)\) unused; \((E)\equiv\) given basis. It was originally expected that left Steinitz rings would turn out to be division rings. Instead the class of perfect local rings turned up, demonstrating also that the intuitive idea that these rings are very suitable for linear algebra and associated geometry is quite correct. The authors of the present paper expand on this idea in a variety of ways. As one example only, if \((A)\equiv\) left, \((B)\equiv\) weakly semi, \((C)\equiv\) finite linearly independent, \((D)\equiv\) finitely generated, \((E)\equiv F\), then the resulting class is extensive as demonstrated (weakly left semi-Steinitz rings) and it has the right properties for finite dimensional linear algebra and geometry. Thus, if it is commutative Noetherian, then it is its own total quotient ring. As a demonstration of the ubiquity of these rings, it is also demonstrated that every reduced \(n\)-dimensional local ring is isomorphic to a subring of a reduced \(n\)-dimensional semi-Steinitz ring (take \((E)\equiv\) given basis instead of \((E)\equiv F)\). Altogether, the paper indicates a rich potential for further development in this entire area, the present paper being an interesting and valuable gathering of results on the way.
reduced \(n\)- dimensional semi-Steinitz ring
scientific article; zbMATH DE number 62235