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Thread: factor rings

  1. #1
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    factor rings

    Draw up an addition and a multiplication table for the factor ring 2
    Z/8Z.What known ring is 2Z/8Z isomorphic

    to?
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  2. #2
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    Draw up an addition and a multiplication table for the factor ring 2
    Z/8Z.What known ring is 2Z/8Z isomorphic to
    If $\displaystyle \phi :R_1\to R_2$ is an isomorphism of commutative rings and $\displaystyle I_1$ is an ideal of $\displaystyle R_1$ then $\displaystyle \phi (I_1) = I_2$ is an ideal of $\displaystyle R_2$ and furthermore $\displaystyle R_1/I_1\simeq R_2/I_2$. Define $\displaystyle \phi: 2\mathbb{Z}\to \mathbb{Z}$ by $\displaystyle \phi(x) = \tfrac{x}{2}$. This is an isomorphism. Let $\displaystyle I_1 = 8\mathbb{Z}$, this is an ideal of $\displaystyle 2\mathbb{Z}$. Note that $\displaystyle I_2 = \phi (I_1) = 4\mathbb{Z}$. Thus, $\displaystyle 2\mathbb{Z}/I_1 \simeq \mathbb{Z}/I_2 = \mathbb{Z}_4$.
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