I’m currently studying Polynomial Rings, but I can’t figure out why they are Rings, not Fields. In the definition of a Field, a Set builds a Commutative Group with Addition and Multiplication. This implies an inverse multiple for every Element in the Set.
The book doesn’t elaborate on this, however. I don’t understand why a Polynomial Ring couldn’t have an inverse multiplicative for every element (at least in the Whole numbers, and it’s already given that it has a neutral element). Could somebody please explain why this can’t be so?
Hint xf(x)=1 in R[x] ⇒ 0=1 in R, by evaluating at x=0
Remark This has a very instructive universal interpretation: if x is a unit in R[x] then so too is every R-algebra element r, as follows by evaluating x f(x)=1 at x=r. Therefore to present a counterexample it suffices to exhibit any nonunit in any R-algebra. A natural choice is the nonunit 0∈R, which yields the above proof.