Group actions and rational ideals

TL;DR

This paper generalizes the theory of rational ideals to arbitrary associative algebras without finiteness assumptions, using the Amitsur-Martindale ring of quotients, and studies the structure of rational ideals under algebraic group actions.

Contribution

It introduces a new framework for rational ideals in general associative algebras and proves a unique orbit correspondence for rational ideals under algebraic group actions.

Findings

Existence and uniqueness of generic rational ideals under G-actions

Closure of G-orbits in the rational spectrum corresponds to rational ideals

Extension of rational ideal theory beyond noetherian and Goldie conditions

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: for every G-rational ideal I of R, the closed subset of the rational spectrum Rat R that is defined by I is the closure of a unique G-orbit in Rat R. Under additional Goldie hypotheses, this was established earlier by Moeglin and Rentschler (in characteristic zero) and by Vonessen (in arbitrary characteristic), answering a question of Dixmier.