Reduced rank and normalizing elements
The author calls a ring R a BPC ring provided every ideal has a finite normalizing set of generators. The paper has three main objectives. Firstly, the author proves that if N is a semiprime ideal of a BPC ring R then the ring R/N contains an Ore set T of regular elements such that the ring of quotients of R/N with respect to T is a finite direct sum of simple rings. Secondly, this fact permits the definition of a bimodule reduced rank analogous to the usual Goldie reduced rank of Noetherian rings. Thirdly an analogue of the Invertible Ideal Theorem of \textit{A. W. Chatters}, \textit{A. W. Goldie}, \textit{C. R. Hajarnavis} and \textit{T. H. Lenagan} [J. Algebra 61, 582-589 (1979; Zbl 0436.16009)] is proved, namely, given a BPC ring R, invertible ideal \(X\neq R\), prime ideal P minimal over X then height \(P\leq 1\). It follows that BPC rings satisfy the descending chain condition on prime ideals.
descending chain condition on prime ideals
scientific article; zbMATH DE number 4049821
scientific article