let R be ring

definition of semiprime ring:

R is semiprime if aRa=0 implies a=0

definition of prime ring:

R is prime if aRb=0 implies a=0 or b=0

from that definition i think semiprime wider than prime (like semiring and ring) thus any prime is semiprime. <== is this true?

if that true, is there any example of semiprime but not prime?