Group actions and rational ideals

Abstract

We develop the theory of rational ideals for arbitrary associative algebras $R$ without assuming the standard finiteness conditions, noetherianness or the Goldie property. The Amitsur–Martindale ring of quotients replaces the classical ring of quotients which underlies the previous definition of rational ideals but is not available in a general setting.

Our main result concerns rational actions of an affine algebraic group $G$ on $R$. Working over an algebraically closed base field, we prove an existence and uniqueness result for generic rational ideals in the sense of Dixmier: for every $G$-rational ideal $I$ of $R$, the closed subset of the rational spectrum $RatR$ that is defined by $I$ is the closure of a unique $G$-orbit in $RatR$. Under additional Goldie hypotheses, this was established earlier by Mœglin and Rentschler (in characteristic $0$) and by Vonessen (in arbitrary characteristic), answering a question of Dixmier.