The center of a prime ring is a domain and the center of a semiprime ring is reduced.

Now I have no evidence to believe that if the center of a semiprime ring R is a domain,

then R has to be a prime ring. So I'm looking for some examples of semiprime rings R

with these properties: R is not prime and the center of R is a domain.

This is not important but it'd be nice if the center of R is not too small.

Thanks for reading