Non-commutative unique factorization domains

Abstract

We introduce a concept of unique factorization for elements in the context of Noetherian rings which are not necessarily commutative. We will call an element p of such a ring R prime if (i) pR = Rp, (ii) pR is a height-1 prime ideal of R, and (iii) R/pR is an integral domain. We define a Noetherian u.f.d. to be a Noetherian integral domain R such that every height-1 prime P of R is principal and R/P is a domain, or equivalently every non-zero element of R is of the form cq, where q is a product of prime elements of R and c has no prime factors. Examples include the Noetherian u.f.d.'s of commutative algebra and also the universal enveloping algebras of solvable Lie algebras. The latter class provides a rich supply of genuinely non-commutative examples.