# example of prime ring and semiprime ring

• Apr 27th 2011, 09:29 AM
wingz00
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?
• Apr 27th 2011, 05:23 PM
NonCommAlg
Quote:

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.

Quote:

if that true, is there any example of semiprime but not prime?
one example of a semiprime ring which is not prime is $\mathbb{Z} \oplus \mathbb{Z}.$ in general, if $R_1, R_2$ are semiprime, then $R_1 \oplus R_2$ is semiprime but not prime.