# 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?