Principal ideal of a unit

Question

I'm taking a Galois Theory course and the course notes begin by revising basic Ring Theory. I came across this:

"In general, $uaR = aR$ for any $u \in R^{\times}$."

(with $R$ being a commutative ring and $R^{\times}$ the set of units of $R$)

I'm having difficulty convincing myself of its truth.

Answer 1

If $(a)\subset R$ is a principal two-sided ideal, then let us consider $(ua)$ for $u$ a unit. The ideal $(ua)$ contains $rua$ for each $r\in R$, and in particular it contains $u^{-1}ua=a$. So $(ua)\supset (a)$, and conversely, $(a)\ni ua\implies (a)\supset uaR$.