Having originated from number theory, the notion of a prime ideal has become central in many different branches of algebra. This talk will focus on the role of prime ideals in the representation theory of noncommutative algebras and the use of group actions as an efficient tool in organizing the spectrum of all prime ideals of a given noncommutative algebra.

The emphasis will be on the case where an affine algebraic group G acts rationally by automorphisms on R. Then the prime spectrum of R is partitioned into "G-strata", each of which can be described in terms of the prime spectrum of a certain *commutative* algebra. Recent work on quantized coordinate algebras R of algebraic varieties has revealed that, in many cases, there is a suitable choice of the acting group G, usually an algebraic torus, such that there are only finitely many G-strata. The resulting finite set tends to carry an interesting combinatorial structure.