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Thread: 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 $\displaystyle R $ be a ring with $\displaystyle m$ elements. Prove that $\displaystyle char(R)$ divides $\displaystyle m$.

    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 $\displaystyle R $ be a ring with $\displaystyle m$ elements. Prove that $\displaystyle char(R)$ divides $\displaystyle m$.

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

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