Let $R$ be a commutative ring with $1$ (we take $R$ not to be a field for this post). Must $R$ contain at least one prime ideal that is not maximal?
The question is equivalent to the following: For a ring $R$ (commutative with 1) is there necessarily a integral domain $S$ which is not a field such that there is a surjective ring homomorphism $R \twoheadrightarrow S$?
I feel that it's not true. Some help would be appreciated. Thanks.