Principal ideal theorem for Noetherian P. I. rings

Abstract

The principal ideal theorem is one of the basic results in the theory of commutative Noetherian rings. A. W. Chattess [l] and P. F. Smith [6] have obtained an analogue of it for those noncommutative Noetherian rings in which the localization at an arbitrary prime ideal can be performed in the usual way. The object of this note is to prove an analogue of the principal ideal theorem for prime Noetherian rings satisfying some polynomial identity. A very neat proof of the principal ideal theorem for commutative Noetherian rings is given by Kaplansky [4, p. 1041. Th e o b vious obstacle in adopting it to the noncommutative situation is that the proof uses the usual localization at a prime ideal whereas no sufficiently nice localization is available at an arbitrary prime ideal in a prime Noetherian ring satisfying a polynomial identity. A closer examination of Kaplansky’s proof reveals that it really hinges on the additivity of the notion of length derived from the JordanHolder theorem; localization is used only to produce a situation in which the Jordan-Holder theorem becomes applicable. We have established in [3] a version of the Jordan-Holder theorem which is applicable to an arbitrary finitely generated module over an FBN-ring (see definition below). It leads to a notion of length which has enough additivity (Theorem 1). Using this notion of length in place of the usual one, we have been able to recast Kaplansky’s proof in such a way that it works for prime Noetherian rings satisfying some polynomial identity.