# Commutative Rings, Subrings and Submodules

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.

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Proof:

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

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