In the eighties, Chatters and Jordan have developed the notion of noncommutative noetherian unique factorisation domain. In this talk, I will recall this notion and show that the algebra of generic quantum matrices which is a noncommuative deformation of the variety of matrices is actually a noncommutative noetherian UFD. To achieve
this aim, we will generalise to the noncommutative world some results coming from the commutative world, and use the rich combinatorics of the algebra of generic quantum matrices. This is joint work with Tom Lenagan and Laurent Rigal.