It's seems a bit absurd to me to go further than non-fields, because fields have no non-zero non-units by definition. That is a nicer argument than the one I h

It's seems a bit absurd to me to go further than non-fields, because fields have no non-zero non-units by definition. That is a nicer argument than the one I had come up with though, which used the observation that all such rings must be quotients of the polynomial ring Z[x].