Abstract.

We study rational actions of a linear algebraic group G on an algebra V, and the induced actions on Rat(V), the spectrum of rational ideals of V (a subset of Spec(V) which often includes all primitive ideals).  This work extends results of Moeglin and Rentschler to prime characteristic, often also relaxing their finiteness assumptions on V.  In particular, we relate properties of a rational ideal J and its orb (J:G), which is the intersection of all ideals g(J) as g runs over the elements of G.  The rational ideals of V containing the orb of J are precisely those in the Zariski-closure X of the orbit of J in Rat(V). The G-stratum of J consists of all rational ideals in X whose orbit is dense in X (i.e., whose orb is equal to the orb of J).  We show that the G-stratum of a rational ideal consists of exactly one G-orbit, and that rational ideals are maximal in their strata in a strong sense.  These results are useful for studying prime and primitive spectra of certain algebras, cf. recent work by Goodearl and Letzter. We further show that the orbit of J is open in its closure in Rat(V), provided that J is locally closed. Among other results, we show that the semiprime ideal (J:G) is Goldie, and we relate the uniform and Gelfand-Kirillov dimensions of V/J and V/(J:G).