Example of the non-commutative ring with the set of units are commutative

Question

I was looking for an Example of the non-commutative ring with the set of units are commutative. it will be a great help. Thanks in advance.

Answer 1

Take the ring of noncommutative polynomials $R\langle X_1, X_2\rangle$ over any commutative ring $R$. The units will be the same as in R.

NB: this is a special case of Mike Debellevue's answer, with $M$ being the free monoid on two generators.

Answer 2

For any monoid $M$ with no units and trivial center, and any commutative ring $R$, the monoid ring $R[M]$ will have the same units as $R$.