Polycyclic group rings and unique factorisation rings

Abstract

The theory of unique factorisation in commutative rings has recently been extended to noncommutative Noetherian rings in several ways. Recall that an element x of a ring R is said to be normalif xR = Rx. We will say that an element p of a ring R is (completely) prime if p is a nonzero normal element of R and pR is a (completely) prime ideal. In [2], a Noetherian unique factorisation domain (or Noetherian UFD) is defined to be a Noetherian domain in which every nonzero prime ideal contains a completely prime element: this concept is generalised in [4], where a Noetherian unique factorisation ring(or Noetherian UFR) is defined as a prime Noetherian ring in which every nonzero prime ideal contains a nonzero prime element; note that it follows from the noncommutative version of the Principal Ideal Theorem that in a Noetherian UFR, if pis a prime element then the height of the prime ideal pR must be equal to 1. Surprisingly many classes of noncommutative Noetherian rings are known to be UFDs or UFRs: see [2] and [4] for details. This theory has recently been extended still further, to cover certain classes of non-Noetherian rings: see [3].