Let $R$ be a commutative, unital ring and let $I$ be an ideal of $R.$ Prove that if $I$ is maximal then $R/I$ is a field.

Question

Let $R$ be a commutative, unital ring and let $I$ be an ideal of $R.$ Prove that if $I$ is maximal then $R/I$ is a field.

This problem is probably trivial for most of you abstract algebra fans out there, but remember that I'm new to the field, so please don't use some next-level math that I can't understand at all. I think this problem could be solved using next-level math very quickly though.

For this problem, I know that fields are commutative, unital, and every nonzero element is a unit/invertible. Also, $I$ is maximal wrt subset, so if there is an ideal that contains $I,$ then it must be $R.$

score 1 · Accepted Answer · answered Nov 03 '19 at 02:38

1

Hint: If $a\notin I$, then $I+Ra$ contains $1$.
Use this to show that $a+I$ has a multiplicative inverse in the quotient ring.

answered Nov 03 '19 at 02:38

Berci

92,013

By $Ra,$ do you mean $\langle a\rangle$? That is, does $Ra$ refer to the principal ideal generated by $a$? – Nov 07 '19 at 01:56
Yes, $Ra$ denotes ${ra:r\in R}$. – Berci Nov 07 '19 at 07:18

Let $R$ be a commutative, unital ring and let $I$ be an ideal of $R.$ Prove that if $I$ is maximal then $R/I$ is a field.

1 Answers1

Linked