Prime ideal and maximal

Quote:

Originally Posted by tttcomrader

If R is a finite commutative ring with unity, prove that every prime ideal of R is a maximal.

Proof. Suppose that <p> is a prime ideal of R, then p is a prime. Let p | ab, we have p|a or p|b. Let $<p> \subset J \subset R$ , I wish to show that J = R.

What can I use here?

Let $\mathfrak{n}$ be an ideal. If $\mathfrak{n}$ is a prime ideal then $R/\mathfrak{n}$ is an integral domain. But it is a finite integral domain. Thus, $R/\mathfrak{n}$ is a field, thus, $\mathfrak{n}$ must be a maximal ideal. Q.E.D.