All prime ideals are maximal - Counterexample

Question

I would like to know of some simple counter examples to the statement "ALL prime ideals are maximal" I say counter examples because I think the statement isn't true.

Answer 1

First you might want to recall that in a domain $\{0\}$ is a prime ideal. Thus, if you have any non-zero prime ideals you have your examples.

Perhaps, you however meant non-zero prime ideals that are not maximal. This is also not difficult to find.

Take a polynomial ring, over som field, in several variables like $k[x,y,z]$. The ideals generated by $\{x\}$, by $\{x,y\}$ and $\{x,y,z\}$ are all prime and only the last one is maximal.

Answer 2

Easy counterexample: consider $\langle x \rangle \subset \Bbb Z[x]$. Alternatively (following quid's idea), consider $\langle x \rangle \subset \Bbb C[x,y]$.

Remember that an ideal is maximal in $R$ if and only if $R/I$ is a field, and that an ideal is prime in $R$ if and only if (it is non the entire ring and) $R/I$ is an integral domain.

Answer 3

An ideal of $\mathbb{C}[z_1, \ldots, z_n]$ can be thought of as a system of polynomial equations, and as such can be associated to its set of solutions in $\mathbb{C}^n$.

Prime ideals correspond to irreducible such loci, and by Hilbert's nullstellensatz there's a one-to-one, order-reversing correspondence between prime ideals and irreducible subvarieties of $\mathbb{C}^n$.

Consequently, a maximal ideal in $\mathbb{C}[z_1, \ldots, z_n]$ is simply the ideal consisting of all polynomials vanishing at a single point. On the other hand, a prime ideal is the ideal of polynomials vanishing on some irreducible subvariety, say a curve or surface in $\mathbb{C}^n$.

So, to get an example, just take your favorite curve or surface or whatever in $\mathbb{C}^n$ and take the ideal of polynomials that vanish on it.

Answer 4

I dont give example; but I want to say where you can find them:
Let $R$ be a Noetherian ring. then $R$ is an Artinian ring iff "ALL prime ideals are maximal". So in any noetherian ring which is not Artinian, (in other words: any noetherian ring of positive dimension), there will be prime ideals which are not maximal.
So you will find them for example in $\Bbb Z$ and $k[x_1,\cdots , x_t]$. think about minimal prime ideals