So I was reviewing for my algebra exam, and one of the review questions was: if a commutative ring has only two ideals is it a field?
I've seen a couple proofs, but the proof I constructed seems like a simpler, although not necessarily more intuitive proof:
Let R be a commutative ring with two ideals.
1) consider R X R.
2) let I = {(a,0) such that a is in R}
3) I is maximal in R X R since R's only ideals are (1), and (0).
4) let F = R X R / I
5) F = {(0,b) + I such that b is in R}
6) F is a field since R X R is a commutative ring, and I is a maximal ideal in this ring.
7) F is isomorphic to R by f : R X R / I -> R, f((0,b) + I) = b
8) since F is a field and R is isomorphic to F, R is a field.