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    Commutative Rings, Subrings and Submodules

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    Problem: I need to show that (i) leads to (ii), then (ii) leads to (iii):

    Let S be a commutative ring, R be a subring in S and x be an element from S.
    Show that the following are equivalent:
    (i) There exist from R where such that
    ;
    In other words x is a root of normalized polynomial over R.
    (ii) Submodule R - module over S which generated by family is finitely generated.
    (iii) There exist a subring in S such that and is finitely generated module over R.

    Please see the attached file for the fully formatted problems.

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    https://brainmass.com/math/ring-theory/commutative-rings-subrings-submodules-117590

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    Proof:
    (i)=>(ii): I claim that the R-module is finitely generated. Actually, ...

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