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Let $R$ be a ring and $S\supset R$ an extension. Let $x\in S$. I have the following statement which are equivalent

  • $x$ is integral over $R$
  • $R[x]$ is finitely generated as an $R$-module
  • There exists a faithful $R[x]$ module finitely generated as an $R$-module

Now my question is what does $R[x]$ mean? I mean we only had $R[X]$ and this is the polynomial ring over $R$ so the ring containing all polynomials in one variable $X$ with coefficients in $R$. But in $R[x]$ the $x$ is an element and not just a variable so I don't see what this means. Could maybe someone explain this?

user1294729
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    It is the subring of $S$ generated by the subring $R$ and the element $x$, i.e. smallest subring containing these. Equivalently, it is the subring of all elements of $S$ of the form $p(x)$ for any $p \in R[X]$, hence the notation $R[x]$. – lisyarus Jun 15 '22 at 06:12
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    @lisyarus ah so for example if I take $R=\Bbb{Z}$ and then consider $\Bbb{Z}[\sqrt{5}]$ then this is the ring of all elements of $p(\sqrt{5})$ where $p\in \Bbb{Z}[X]$? – user1294729 Jun 15 '22 at 06:15
  • It's a ring-adjunction - see the linked dupe (and its links). – Bill Dubuque Jun 15 '22 at 07:35

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$R[x] = \{ \ f(x)\ | \ f \in R[X] \ \}$, where $x \in S$ and $R[X]$ is the ring of polynomials with coefficients in $R$

just to clear the air, if $\alpha \in S$ then $R[\alpha]$ is exactly the evaluation of the polynomials $f \in R[X]$ at $\alpha$

codehumor
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