On weakly semi-Steinitz rings (Q2785922)

All rings \(A\) have an identity. Steinitz rings were originally discovered in the context of ``linear algebra'' via the requirement that the Steinitz Replacement Theorem should hold in its unlimited version. Various generalizations, e.g., cyclic rings and central perfect rings have already been looked at to advantage. In this interesting paper the authors identify those commutative rings for which every finite linearly independent subset of a finitely generated free \(A\)-module \(F\) can be extended to a basis. Among various characterizations, including theorem 1, the most straightforward one is the following (theorem 2): the commutative ring \(A\) is weakly semi-Steinitz if every regular element of \(A\) is invertible and if any two maximal linearly independent subsets of a free \(A\)-module have the same cardinality. This permits for a sequence of observations culminating in the observation that \(T(A[X])\), the total quotient ring of the polynomial ring \(A[X]\) is weakly semi-Steinitz for any commutative ring \(A\). Other applications of interest also show that it is easy for commutative rings to have nice semi-Steinitz overrings which maintain maximal ideal spaces.NEWLINENEWLINEFor the entire collection see [Zbl 0855.00015].