Explore BrainMass

# Local ring proof

Not what you're looking for? Search our solutions OR ask your own Custom question.

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

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.

https://brainmass.com/math/ring-theory/local-ring-proof-241297

#### Solution Preview

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 ...

#### Solution Summary

This provides an example of proving a ring is local.

\$2.49