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.