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Control subgroups and birational extensions of graded rings. (English) Zbl 0932.16042

Let \(R\) be a strongly \(G\)-graded ring and \(H\) a normal subgroup of \(G\). The author proves that \(R^{(H)}\subseteq R\) is a Zariski extension if and only if the filter \({\mathcal L}(R-P)\) is controlled by \(H\) for any prime ideal \(P\) in an open set of the Zariski topology on \(R\). As an application certain ideals of \(R\) and \(R^{(H)}\) are related up to radical.

MSC:

16W50 Graded rings and modules (associative rings and algebras)
16S34 Group rings
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
16D25 Ideals in associative algebras
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