In this paper we study rings and bimodules with no known one-sided chain conditions, but whose (two-sided) ideals and subbimodules are ‘nicely’ generated. We define bi-noetherian polycentral (BPC) and bi-noetherian polynormal (BPN) rings and bimodules, large classes of (almost always) non-noetherian objects, and put on record the basic facts about them. Any BPC ring is a BPN ring. In the case of rings we reduce their properties to properties of the prime ideals, and study the d.c.c. on (two-sided) ideals. We define both the artinian and bi-artinian radicals of a BPN ring, and use them to show that for BPN rings the intersections of the powers of both the Brown-McCoy and the Jacobson radicals are zero.