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Say $R$ is a commutative ring. Does there exists a subring of $R[x]$ that is isomorphic to $R$?

My approach would be to define the subring of $R[x]$ that generates $R$. Any thoughts?

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Constant polynomials. – Brian M. Scott Nov 17 '12 at 10:23
up vote 7 down vote accepted

The ring of constant polynomials in $R[x]$ is isomorphic to $R$

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But the crux of the matter is to prove that! See my answer for one easy way. – Bill Dubuque Nov 17 '12 at 15:35

Hint $ $ The evaluation hom $\rm\:f(x)\to f(0)\:$ is $1$-$1$, onto, restricted to polynomials of degree zero.

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ok. +1 for quick way. – Amr Nov 17 '12 at 15:53
@Bil: How is the function 1-1, since I can take $f(x)$ and $g(x)$ to be two different functions in $R[x]$ with the same constant term and the value of $f(0)$ and $g(0)$ will still be same. So, I don't understand how will it be 1-1? – MUH Jun 29 at 9:51
@MUH the claim is restricted to polynomials of degree zero (constants). – Bill Dubuque Jun 29 at 13:06

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