Is Z7[x] Isomorphic to Z?

[jouiswalker]

Homework Statement

Let R = Z7[x]. Show that R is not isomorphic to Z.

Homework Equations

The Attempt at a Solution

One of the necessary conditions for an isomorphism f is that f be one to one. So consider 8x in Z. f(8x) = x, f(1x) = x. So f cannot be an isomorphism. I'm clearly missing something though, since this seems a bit too easy.

 

[jouiswalker]

Actually nevermind, it is that easy. There's a hint hidden that basically says "an isomorphism has to be one to one..."