Can a Unique Factorisation Domain be non-commutative? The definition that our lecturer gave us for Unique Factorisation Domains is:

An integral domain $R$ is called a Unique Factorisation Domain (UFD) if every non-zero non-unit element of $R$ can be written as a product of irreducible elements and this product is unique up to order of the factors and multiplication by units.

If multiplication in this integral domain is non-commutative, then if $x, a, b \in R$ and $x = ab = ba$, do these count as different factorisations and mean that $R$ can't be a UFD?

By the usual definition, an integral domain is commutative.
