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Math Help - Characteristic and size of a Ring

  1. #1
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    Acolman, Mexico
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    Characteristic and size of a Ring

    I need help with the following problem:

    Let R be a ring with m elements. Prove that char(R) divides m.

    Thanks in advance.
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  2. #2
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    Quote Originally Posted by akolman View Post
    I need help with the following problem:

    Let R be a ring with m elements. Prove that char(R) divides m.

    Thanks in advance.
    (R,+) is a group of order m.. let d=\text{char}(R) and m=kd + r, where 0 \leq r < d. suppose that 0 < r < d. by Lagrange theorem for finite groups m \cdot 1_R=0. thus 0=(kd + r) \cdot 1_R = r \cdot 1_R,

    which contradicts the definition of "charateristic" because our assumption was that 0 < r < d. hence 0 < r < d is impossible and so we must have r=0, i.e. m=kd.
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