The discussion revolves around demonstrating that the rings Z[x] and Z are not isomorphic. Participants are exploring structural properties and characteristics that differentiate these two rings.
The discussion is ongoing, with participants raising various approaches to the problem. There is an exploration of the implications of assuming an isomorphism and the nature of ideals in Z. Some participants suggest considering the fact that Z[x] is not a principal ideal domain (PID) as a potential avenue for differentiation.
Participants note the challenge of finding specific structural differences between Z[x] and Z, as well as the constraints of the homework prompt which requires a demonstration of non-isomorphism.
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}##