I am for the first time teaching Algebra 1 practicals. With students, we were proving that
If T is a field, then the ring of polynomials T[x] is Euclidean
Now I know that the opposite is not true, because if T is not a field, T[x] is not even P.I.D. I would like to prove this statement in class, but I do not know any accessible proof (my proof is by investigating whether the factor-rings of R[x] are Artinian ).
One can easily show, that if R has a non-invertible element $a\neq 0$, then the Euclidean norm from T[x] (degree of polynomial plus 1) wouldn't work as shown by the division of polynomials $x:ax$.
But I do not know how to prove that any other norm will also fail to be Euclidean. I have never actually constructed such proof. When I was disproving Euclideaness I always disproved some weaker property (e.g. I proved that the ring is not even P.I.D.).
I am not sure whether elementary proof exists, but I feel that there should be one. Do you have any idea?
