## Rings with a finite number of non-units

Posted: September 17, 2010 in Elementary Algebra; Problems & Solutions, Rings and Modules
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Problem. Let $R$ be an infinite ring with 1 and let $U(R)$ be the set of units of $R.$ Prove that if $R \setminus U(R)$ is finite, then $R$ is a division ring.

Solution. Suppose, to the contrary, that there exists some $0 \neq x \in R \setminus U(R).$ First note that if $I \neq R$ is a left or right ideal of $R,$ then $I$ is finite because otherwise $I \cap U(R) \neq \emptyset$ and so $I=R.$ Therefore $Rx$ and $xR$ cannot both be infinite. Suppose that $Rx$ is finite and let $I=\{r \in R: \ rx = 0 \}.$ Then $I$ is a left ideal of $R$ and $I \neq R$ because $x \neq 0.$ Hence $I$ is finite. Now the $R$-module isomorphism $R/I \cong Rx$ implies that $R$ is finite. Contradiction!