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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.

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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.

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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 $R$.

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  • $\begingroup$ Nice! My example is a special case when $M$ is the free monoid on two generators. $\endgroup$ – lisyarus Jun 1 at 20:49

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