A well known equivalent of the Axiom of Choice is Krull's Maximal Ideal Theorem (1929): if $I$ is a proper ideal of a ring $R$ (with unity), then $R$ has a maximal ideal containing $I$. The proof is easy with Zorn's Lemma. The converse, Krull implies Zorn, is due to Hodges (1979).

A ring with a unique maximal ideal is said to be local. A standard homework problem is this: Show that if $R$ is a local ring with maximal ideal $M$, then every element outside of $M$ is a unit. The "usual" solution is as follows: Let $x$ be a nonunit and let $I$ be the ideal generated by $x$. By Krull, $I$ is contained in a maximal ideal which, by locality, must be $M$.

Since the solution relies on AC, a natural question arises:

Is the homework problem itself equivalent to AC? More precisely, assume that in every local ring, every element outside of the unique maximal ideal is a unit. Does this imply AC?

If the answer is no, is there a proof of the homework problem which does not rely on AC?