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Thread: Subring of the complex

  1. #1
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    Subring of the complex

    Let A the subring of the complex $\displaystyle A=\{ a + b \sqrt{2}: a,b \in \mathbb{Z}\}$

    a) Prove that $\displaystyle A$ is a $\displaystyle \mathbb{Z}$-module and an $\displaystyle A$-module, with the product as action

    b) Prove that the function $\displaystyle a + b \sqrt{2} \rightarrow a+b$ is a homomorphism of $\displaystyle \mathbb{Z}$-modules from $\displaystyle A$ to $\displaystyle A$ but isnīt a homomorphism of $\displaystyle A$-modules

    c) Prove that, as $\displaystyle \mathbb{Z}$-module, A is isomorphic to $\displaystyle \mathbb{Z} \oplus \mathbb{Z}$

    thanks!!
    Last edited by roporte; Nov 7th 2008 at 08:29 PM. Reason: mistake
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