Hypercentral ideals, noetherian modules, and a theorem of Stroud

Abstract

Received April 9, 1973 1. INTRODUCTION An ideal which can be generated by elements belonging to the centre of a ring R is called a central ideal of R: here and in what follows “ring” always means a ring with unity. More generally, an ideal I of R is said to be hypeycevztral if there exists an ascending central seties of ideals of R from 0 to 1, that is to say, a chain of ideals 0 = I, < II < ... I, <I&+1 < ... I, = I, such that &+i/& is a central ideal of R/I, and I,, = &,,Ia for all ordinals a < L and all limit ordinals h < 1. The least ordinal L which is the length of such a series is called the height of I. Recently Roseblade [5] proved that every ideal of the integral group ring ZG of a group G is hypercentral if and only if G is a hypercentral group, i.e., G coincides with a (possibly trans- finite) term of its upper central series. Hypercentral ideals which have finite height will be called po&central, but the reader should note that Roseblade uses this term in a narrower sense. If M is a (right unitary) R-module and S is a nonempty subset of R, let *S stand for the additive subgroup of all a in n/I such that as = 0 for all s E S. Our first object is to prove the following theorem. THEOREM 1. Let R be a ving and lM a noetheviavz R-vvwdule. Then there exists a positive integer n such that MP