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 , I wish to show that J = R.
What can I use here?
Let be an ideal. If is a prime ideal then is an integral domain. But it is a finite integral domain. Thus, is a field, thus, must be a maximal ideal. Q.E.D.