A ring R is semiprime if 0 is a semiprime ideal.
By correspondence, a kernel is prime/semiprime iff its quotient ring is same.
The direct product of prime rings fails to be prime. Multiply P,0 by 0,P to get 0,0 - whence the zero ideal is not prime. In contrast, the direct product of semiprime rings is semiprime. The xRx test passes, because it passes per component.