PRIME IDEALS IN SKEW AND Q-SKEW POLYNOMIAL-RINGS

New methods axe developed to describe prime ideals in skew polynomial rings S = R[y; tau, delta], for automorphims tau and tau-derivations d elta of a noetherian coefficient ring R. A complete description is giv en under the additional assumptions that R is an algebra over a field k on which tau and delta act trivially and that tau-1 deltatau = qdelt a for some nonzero element q is-an-element-of k. These hypotheses abst ract behavior found in many quantum algebras, such as q-Weyl algebras and coordinate rings of quantum matrices, and specific examples along these lines are considered in detail. Finally, we prove, for a general class of n-fold iterated skew polynomial extensions (including the 1- fold extensions of the first sentence), that there cannot exist chains of more than n + 1 prime ideals all lying over a common prime ideal o f the coefficient ring.