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Math Help - Isomorphic Ring

  1. #1
    Senior Member vincisonfire's Avatar
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    Isomorphic Ring

    Prove that no two of the following rings are isomorphic:
    (a) R R R R (with addition and multiplication given coordinate by coordinate);
    (b) M2 (R) (2 x 2 matrices with real coefficient);
    (c) The ring H of real quaternions (are they simply integers?).
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    Quote Originally Posted by vincisonfire View Post
    (a) R R R R (with addition and multiplication given coordinate by coordinate);
    (b) M2 (R) (2 x 2 matrices with real coefficient);
    (c) The ring H of real quaternions (are they simply integers?).
    1) Ring (a) is commutive but ring (b) and (c) are not.
    2) Ring (c) is a strictly skew field but (b) is not.
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    Senior Member vincisonfire's Avatar
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    For number two you mean :
    - There exists an inverse with respect to multiplication for ALL quaternions.
    - There exists an inverse with respect to multiplication for SOME real 2x2 matrices.
    That's it?
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    Quote Originally Posted by vincisonfire View Post
    For number two you mean :
    - There exists an inverse with respect to multiplication for ALL quaternions.
    - There exists an inverse with respect to multiplication for SOME real 2x2 matrices.
    That's it?
    Exactly. In M_{2\times 2}(\mathbb{R}) we have that not all non-zero elements have inverse. Whiles with \mathbb{H} (quaternions) all non-zero elements have inverses.
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