Re: problem with rings of polynomials

[Bill Dubuque]

THEOREM Let F in R[X] be a poly over a commutative ring R.
If F is a zero-divisor then rF = 0 for some nonzero r in R.

PROOF. Suppose not. Choose G != 0 of min deg with FG = 0.

Write F = a +...+ f X^k +...+ c X^m
G = b +...+ g X^n, where g != 0 and

where f is the highest deg coef of F with fG != 0
(such f exists else Fg = 0 contra supposition).

Then FG = (a +...+ f X^k)(b +...+ g X^n) = 0

Thus fg = 0 => deg(fG) < n & F fG = 0

contra choice of G of min deg. QED

Alternatively it's an immediate corollary of Gauss' Lemma
(Dedekind-Mertens) or related results.

-Bill Dubuque