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Local ring proof

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A local ring is commutative with identity which has a unique maximal ideal. Prove that R is local if and only if the non-units of R form an ideal.

I need a detailed, rigorous proof of this with explanation of the steps so I can learn. Thank you.

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This provides an example of proving a ring is local.

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Proof:
"<=": Suppose M is the set of all non-units of R and M form an ideal.
First, I claim that M is a maximal ideal of R. Because if M is contained by another ideal I, then I contains some
element x that does not belong to M, then x must be a unit. So we can find some y in R, such that yx = 1. Since
I is an ideal, then yx = 1 is in I, thus I = R. Therefore, M is a maximal ...

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