TITLE: Prove: If $x$ is not a left zero divisor and $xy=1$, then $yx=1$.
QUESTION [8 upvotes]: Let $R$ be a ring with unity, and let $x,y \in R$ with $xy = 1$. Assume
that $x$ is not a left zero-divisor. Prove that $x$ is a unit.
$$xy = 1$$
$$\begin{gather}
xyx = x \\
xyx - x = 0 \\
x(yx-1) = 0
\end{gather}$$
Since $x$ is not a left zero divisor, $yx-1 = 0$.
Thus $yx=1$.
As $xy=1$ as well, $x$ is a unit. 
Is this right?

REPLY [3 votes]: Your solution is correct.
(I'm posting this CW answer to get this off the unanswered queue.)