Dubrovin valuation skew group rings
In 1982, \textit{N.~I.~Dubrovin} [see Tr. Mosk. Mat. O.-va 45, 265-280 (1982; Zbl 0531.16011)] introduced a noncommutative valuation ring, now known as a Dubrovin valuation ring, and he showed that \(R\) is such a ring if and only if it is a semihereditary local order in a simple Artinian ring. Using this result, the paper under review determines when certain skew group rings are Dubrovin valuation rings. Let \(R\) be a commutative ring which is an order in an Artinian ring and let \(G\) be a finite group. Let \(t\colon G\to\Aut(R)\) be an action of \(G\) on \(R\) and let \(R_tG\) be the corresponding skew group ring. One interesting result here is that \(R_tG\) is a Dubrovin valuation ring if and only if the crossed products \(R_t^\alpha G\) are Dubrovin valuation rings for each \(t\)-cocycle \(\alpha\).
scientific article; zbMATH DE number 1885834