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.© BrainMass Inc. brainmass.com March 21, 2019, 6:00 pm ad1c9bdddf
"<=": 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 ...
This provides an example of proving a ring is local.