Left inverse implies right inverse in a finite ring

Question

Let $R$ be a finite ring with identity $1$, and assume $\exists x,y\in R$ such that $ xy=1$. How can I show it implies $yx=1$?

Answer 1

Hint: $xy=1$ implies that left multiplication by $y$ is one-to-one. Can you draw a conclusion whether or not there is a $z$ such that $yz=1$?

If so, you can complete the argument by showing that $z=x$.

Answer 2

Hint $\ $ As often occurs, this result on numbers is a special case of a result on functions. namely, consider $\rm\:x,y\:$ as left-multiplication maps $\rm\:f(r) = xr,\ g(r) = yr,\:$ then apply the following

Lemma $\rm\ fg = 1\ \Rightarrow\ gf = 1\ $ for maps $\rm\:f,g\:$ on a finite set $\rm\:R.$

$\rm(1)\ \ \ fg = 1\ \Rightarrow\ g\ is\ 1\!-\!1\:$ by $\rm\:f\:$ of $\rm\:g(a) = g(b)\ \Rightarrow\ a = b $

$\rm(2)\ \ \ g\ is\ 1\!-\!1\ \Rightarrow\ g\:$ is onto, since $\rm\:R\:$ is finite

$\rm(3)\ \ \ g\ is\ onto\ \Rightarrow\ gf = 1\:$ by $\rm\ a = g(b) = g(fg(b)) = gf(a)$

Remark $\ $ In fact we may view the ring as the set of such maps (left-regular representation), where the elements of $\rm\:R\:$ are essentially viewed as $1$-dimensional matrices. Then the above is analogous to a well-known result about matrices, e.g. see my post here where I prove $\rm\ AB = I\:\Rightarrow\; BA = 1,\:$ or, equivalently, $\rm\:B\:$ injective $\rm \Rightarrow$ $\rm\: B\:$ surjective, by exploiting the pigeonhole principle. See also other posts in that thread which clarify the fundamental role played by the pigeonhole principle. See also this question on Dedekind-finite rings, i.e. rings where $\rm\:xy = 1\:\Rightarrow\: yx = 1.$