| We describe those ideals of an arbitrary associative unital ring R which extend to prime ideals in three noncommutative extensions. The first extension is the polynomial extension S = R[x; sigma] invented by Ore. In the case when sigma is surjective, we completely characterize those ideals I of R for which IS is a prime ideal. We do the same for the closely related skew Laurent-polynomial extensions R[x, x-1; sigma]. Our results include specialization to Noetherian base rings, when these ideals are closely related to prime ideals of R. In addition, we generalize a well known result from commutative algebra regarding the homogeneity of associated prime ideals in Z -graded rings. The last extensions, invented by Bavula, are known as generalized Weyl algebras. These extensions are closely related to the previous extensions. We characterize those ideals of R which extend to ideals of A in several surprising ways.; The material appearing in Chapter II has been previously published under the same title in the Journal of Algebra. |
