The rings Z[x] and Z are not isomorphic due to fundamental structural differences. While it is established that the quotient ring Z[x]/(x) is isomorphic to Z, Z[x] itself is not a Principal Ideal Domain (PID), which Z is. The existence of ideals such as I = ⟨2, 1+x⟩ in Z[x] demonstrates that not all ideals can be generated by a single element, a property that Z possesses. Therefore, the non-isomorphism is confirmed through the analysis of their ideal structures.
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I meant to write ##\mathbb{Z}[x]/(x) \cong \mathbb{Z}##fresh_42 said:##\mathbb{Z}/(x) \cong \mathbb{Z}##?
Another idea is to use the fact that ##\mathbb{Z}[x]## is no PID, e.g. ##I:=\langle 2, 1+x \rangle##.Mr Davis 97 said:Homework Statement
Show that the rings Z[x] and Z are not isomorphic
Homework Equations
The Attempt at a Solution
I want to show that these are not isomorphic. The thing is that I already know that ##\mathbb{Z}/(x) \cong \mathbb{Z}##, but for some reason I can't find specific structural properties of ##\mathbb{Z}[x]## that are different than ##\mathbb{Z}##