TITLE: Can the following be done without zorn's lemma
QUESTION [5 upvotes]: Prove that $R$ is local ring with maximal ideal $M$ if and only if every element of $R\backslash M$ is a unit.
I understand that this is a easy consequence of zorn's lemma. But I'm wondering if this can be done without zorn's lemma.
Any help or insight is deeply appreciated.

REPLY [10 votes]: As noted in comments, Zorn's Lemma is known to be equivalent to the statement that every (nonzero, unital) commutative ring has a maximal ideal.
Suppose $S$ is a commutative ring without a maximal ideal, and $k$ a field. Then $R=k\times S$ has a unique maximal ideal $M=0\times S$, but not every element of $R\setminus M$ is a unit.
So the full strength of Zorn's Lemma is needed to prove that if a commutative ring $R$ has a unique maximal ideal $M$, then every element of $R\setminus M$ is a unit.