Integral domain without prime elements which is not a field

Question

I was asked whether I knew a ring without prime elements which is not a field. The first thing I thought of was the Cartesian product of fields with component-wise addition and multiplication. But now I am looking for a ring which does not have any zero-divisors. I could not think of one.

So does anyone know an integral domain (commutative ring without zero divisors) which is not a field and has no prime elements?

My first idea was adjoining elements to a known ring such as $\mathbb{Z}[\alpha]$. But then the problem is that I always find some prime in $\mathbb{Z}$, which is also a prime in this new ring.

Answer 1

Take the ring of power series $R=k[[X,X^{1/2},X^{1/3},\ldots]]$.

If $f\in R$ is non-invertible, then $f = uX^{1/n}$ for some unit $u$ and positive integer $n$. But $X^{1/n}$ is not prime, so $f$ is not prime.

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For a noetherian example, take the ring $k[[X^2,X^3]]$, or $k[X^2,X^3]_{(X^2,X^3)}$.

More generally, if $C$ is an irreducible curve with $y\in C$, then the local ring $R=\mathcal{O}_{C,y}$ will be a noetherian domain with a unique nonzero prime ideal $P$. If $y$ is a singularity, then $P$ will not be principal.