Group actions and rational ideals


Algebra and Number Theory 2, 467-499 (2008); 
Abstract: We develop the theory of rational ideals for arbitrary associative algebras R (with 1) 
over some base field k . Specifically, we define a prime ideal  P  of  R  to be rational if the 
extended centroid  C(R/P), in the sense of Martindale,  is equal to the base field  k. If a group  
G  acts by k-algebra automorphisms on  R  then a G-prime ideal  I  of  R  will be called 
G-rational if the algebra of G-invariants  C(R/P)G  is equal to  k.  We show that, for any rational 
ideal  P  of  R, the intersection of the ideals in the G-orbit of  P is G-rational.
Our main result concerns rational actions of an affine algebraic group  G  on  R. Assuming  k  to 
be algebraically closed, we show:  (1) for every G-rational ideal  I  of  R  there exists a rational 
ideal   P  of  R  such that  I  is the intersection of the ideals in the G-orbit of  P, and   (2) the G-orbit 
of  P  is uniquely determined by  I.
Under additional finiteness conditions, this result is due to Moeglin and Rentschler (in characteristic 0) 
and to Vonessen (in arbitrary characteristics).


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