We determine when a generalized down-up algebra is a Noetherian unique factorisation domain or a Noetherian unique factorisation ring.the global dimension is always 3). Similarly, in some cases the centre is reduced to the scalars, but in others it can be large, and there are cases in which the generalized down-up algebra is finite over its centre. Other examples of properties which hold in some generalized down-up algebras and do not in others are: being Noetherian, being primitive, having all finite-dimensional modules semisimple, and having a Hopf algebra structure.Generalized down-up algebras have been studied mostly from the point of view of representation theory (see [6], [10], [24], [19], [11] and [29]); their primitive ideals have been determined in [21], [28], [31], [30] and [32]. In this paper we study generalized down-up algebras from the point of view of noncommutative algebraic geometry, namely, we provide a complete classification of those generalized down-up algebras which are (noncommutative) Noetherian unique factorisation rings (resp. domains), as defined by Chatters and Jordan in [12] and [13].An element p of a Noetherian domain R is normal if pR = Rp. In our case, a Noetherian domain R is said to be a unique factorisation ring, Noetherian UFR for short, if R has at least one height one prime ideal, and every height one prime ideal is generated by a normal element. If, in addition, every height one prime factor of R is a domain, then R is called a unique factorisation domain, Noetherian UFD for short. As well as the usual commutative Noetherian UFDs, examples of Noetherian UFDs include certain group algebras of polycyclic-by-finite groups [8] and various quantum algebras [26,25] such as quantised coordinate rings of semisimple groups. Unfortunately, the notion of a Noetherian UFD is not closed under polynomial extensions. To the opposite, the notion of a Noetherian UFR is closed under polynomial extensions. Moreover, Chatters and Jordan proved general results for skew polynomial extensions of the type R[x; σ] and R[x; δ]. The general case of skew polynomial extensions of type R[x; σ, δ] is more intricate and only partial results have been obtained for a class of "quantum" algebras called CGL extensions [26], which includes (generic) quantum matrices, positive parts of quantum enveloping algebras of semisimple Lie algebras, etc.Going back to enveloping algebras, it follows from results of Conze in [14] that, over the complex numbers, the universal enveloping algebra of a finite-dimensional semisimple Lie algebra is a Noetherian UFD, and an analogous result holds for a finite-dimensional solvable Lie algebra, by [12]. It is thus natural to investigate the factorial properties of generalized down-up algebras. Moreover, by considering cases in which the parameters r and s are roots of unity, we hope to get some insight into the behaviour of enveloping algebras over fields of finite characteristic (see [7] and references therein). Indeed, our analysis yields the following result, which shows that, f...
