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Thread: finitely generated module & algebra

  1. #1
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    finitely generated module & algebra

    Let $\displaystyle A$ be finitely generated as a $\displaystyle R$-algebra.

    Then can I say $\displaystyle A$ is finitely generated as a $\displaystyle R$-module?

    I cannot understand some part of definitions about finitely generated...
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  2. #2
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    Quote Originally Posted by Stiger View Post
    Let $\displaystyle A$ be finitely generated as a $\displaystyle R$-algebra.

    Then can I say $\displaystyle A$ is finitely generated as a $\displaystyle R$-module?

    I cannot understand some part of definitions about finitely generated...

    The polynomial algebra $\displaystyle \mathbb{R}[x]$ is finitely generated as an $\displaystyle \mathbb{R}-$algebra, but it is NOT

    a finitely generated $\displaystyle \mathbb{R}-$module...

    ** f.g. algebra = all the elements are finite sums of finite products of some elements in the algebra

    ** f.g. module = all the elements are finite sums AND PRODUCTS BY SCALARS of some elements in the module.

    Tonio
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