How do I prove that $R[x] \cong R[y]$? I understand that $R[n]$ represents the set of all polynomials in $n$ with coefficients from the commutative ring $R$, but I don't know how to even start this problem.
Any help would be great, thank you!
2026-04-06 17:31:51.1775496711
Prove that $R[x] \cong R[y]$
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I will outline an sketch for how you might start thinking about this problem. First, recall that two sets are isomorphic if there exists an isomorphism between the two sets. Consider a polynomial in $R[x]$. How could you map this to $R[y]$? Then show that this mapping is well-defined, injective, surjective, and operation-preserving. Then, you have your explicit isomorphism, so you can say that $R[x]\cong R[y]$.