Thread: example of prime ring and semiprime ring

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?

Originally Posted by wingz00

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?

obviously.

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

one example of a semiprime ring which is not prime is in general, if are semiprime, then is semiprime but not prime.