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Math Help - example of prime ring and semiprime ring

  1. #1
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    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?
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  2. #2
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    Quote Originally Posted by wingz00 View Post
    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 \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.
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