Containment between prime ideals and maximal ideals

In a commutative ring (with unity), is it true that

(a) any maximal ideal is a prime ideal?

(b) any prime ideal is a maximal ideal?

(b) is almost certainly false, because a maximal ideal is a stronger concept than a prime ideal, but I don't know of any example to give.

And I'm not sure about (a).

1 Answer

It may help you to think about the following. An ideal $I$ is maximal iff $R/I$ is a field. An ideal $I$ is prime iff $R/I$ is an integral domain. So, the question boils down to: a) is every field an integral domain? b) can you find an example where $R/I$ is an integral domain but not a field (hint: look around the very first examples of rings and ideals).

• Thanks, Ittay! Looking at $R/I$ makes things very clear. Since it's so clear like this, I have to wonder: Is there a way to see it using the definition of prime ideal and maximal ideal, without going to $R/I$? – Kunal Jan 15 '14 at 2:05
• Yes @Kunal it is possible to give direct proofs without passing to the quotient. It's a good exercise to figure those out. Give it a go! – Ittay Weiss Jan 15 '14 at 2:37