Every field is a local ring

Question

I have a short question about a local ring. I just saw the concept and looked something up on the internet.

I wonderdered if it is true that every field is a local ring?

Because ${0}$ is the only maximal ideal. I'm not sure, because i can't prove that if $R$ is a field then only ${0}$ and $R$ are ideals.

@mathmath Because a ring is local preciesely when it has a unique maximal ideal and a field has a unique maximal ideal. What else is there to say? — rschwieb, Aug 26 '20 at 12:53

score 3 · Accepted Answer · answered Aug 26 '20 at 12:35

3

If $R$ is a field and $I\ne\{0\}$ is an ideal, take $r\in I$, $r\ne0$. Then $$ 1=rr^{-1}\in I $$ so $I=R$.

More generally, a commutative ring $R$ is local if and only if the set of noninvertible elements of $R$ is an ideal.

answered Aug 26 '20 at 12:35

egreg

1 Answers1