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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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Commutative Rings, Subrings and Submodules are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.