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}##
The rings Z[x] and Z are not isomorphic because they have different algebraic structures. Z[x] is a polynomial ring, while Z is an integral domain. This means that the elements in Z[x] are polynomials with coefficients in Z, while the elements in Z are integers. The two rings also have different properties and operations, making it impossible for them to be isomorphic.
Yes, for example, the element x in Z[x] has no counterpart in Z, as it is not an integer. Additionally, the polynomial x+1 in Z[x] has no multiplicative inverse in Z, while every non-zero element in Z has a multiplicative inverse.
Yes, it is possible for Z[x] and Z to have some elements in common. For example, the element 1 is present in both rings. However, the way in which this element is used and the operations performed on it differ in each ring, further highlighting their non-isomorphic nature.
Yes, there are many other rings that are not isomorphic to Z[x]. Some examples include the polynomial ring over a field, such as R[x], and the ring of integers modulo n, denoted by Z/nZ or Zn.
The fact that Z[x] and Z are not isomorphic has significant implications in abstract algebra. It means that these two rings cannot be studied or compared using the same theorems or methods. It also highlights the importance of understanding the distinct properties and structures of different rings, rather than assuming they are all equivalent.