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Pairs of rings invariant under group action

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Abstract

Let \(R\subseteq S\) be a ring extension and \(\mathcal {P}\) be a ring-theoretic property. The pair (RS) is said to be a \(\mathcal {P}\)-pair if, T satisfies \(\mathcal {P}\) for each intermediate ring \(R\subseteq T\subseteq S\). Let G be a subgroup of the automorphism group of S such that R is invariant under the action by G. In this paper we investigate in several cases the transfer of a property \(\mathcal {P}\) from the pair (RS) to \((R^G,S^G)\). For instance, if \(\mathcal {P}:=\) Residaully algebraic, LO, INC, and Valuation, we show that each of these properties pass from (RS) to \((R^G,S^G)\). Additional consequences and applications are given.

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Correspondence to Nabil Zeidi.

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Zeidi, N. Pairs of rings invariant under group action. Beitr Algebra Geom (2020). https://doi.org/10.1007/s13366-020-00484-w

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Keywords

  • Normal pair
  • Valuation domain
  • LO & INC
  • Treed domain
  • Ring of invariants
  • Group action

Mathematics Subject Classification

  • Primary 13B02
  • 13A50
  • Secondary 13A15
  • 13A18
  • 13B22