How to show $u \in R^{x} \implies u \in R[X]^{x}$ for integral domain $(R,+, \times)$

Asked Aug 11 '20 at 09:48

Active Aug 11 '20 at 09:48

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I am studying ring theory and have come across polynomial rings. There is an exercise to show that $$u \in R^{x} \implies u \in R[X]^{x}$$ where $(R,+, \times)$ is an integral domain.

I know that $R^{x}$ is the set of all units in R, which is the set of all elements of $R$ with a multiplicative inverse, and similarly for $R[X]^{x}$. But I am stuck with proving the above statement.

asked Aug 11 '20 at 09:48

mathreads

You should look at how $R$ embeds in $R[X]$. This will provide multiplicative inverses for $u$ in $R[X]$. – Simon Marynissen Aug 11 '20 at 09:52
There are several duplicates, e.g., also this one. – Dietrich Burde Aug 11 '20 at 11:19

How to show $u \in R^{x} \implies u \in R[X]^{x}$ for integral domain $(R,+, \times)$

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