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Let R be a ring. An element p of R is a prime element if pR = Rp is a prime ideal of R. A prime ring R is said to be a Unique Factorisation Ring if every non-zero prime ideal contains a prime element. This paper develops the basic theory of U.F.R.s. We show that every polynomial extension in central indeterminates of a U.F.R. is a U.F.R. We consider in more detail the case when a U.F.R. is either Noetherian or satisfies a polynomial identity. In particular we show that such a ring R is a maximal order, that every height-1 prime ideal of R has a classical localisation in which every two-sided ideal is principal, and that R is the intersection of a left and right Noetherian ring and a simple ring.

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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.

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