# Prime ideals in commutative ring

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.

• How about a field $R$? Or, let $R$ be Artinian. – user714630 May 9 '13 at 14:28
• This answers my question! Thanks. – nigel May 9 '13 at 14:30
• Note also that the Krull dimension is a measure of the lengths of chains of prime ideals, and any zero-dimensional ring has this property. – user714630 May 9 '13 at 14:32
In fact, any commutative Artinian ring will do, since any prime ideal $\mathfrak p$ contains a product of maximal ideals. This implies $\mathfrak p$ is itself a maximal ideal.