Show that if R has no zero divisors then R[x] also has no zero divisors.

Printable View

- Dec 25th 2012, 04:34 PMBernhardPolynomial rings - zero divisors in R and r[x]
Show that if R has no zero divisors then R[x] also has no zero divisors.

- Dec 25th 2012, 05:11 PMBernhardRe: Polynomial rings - zero divisors in R and r[x]
Thinking about this problem ...

Let

and

Then

where

But cannot equal zero since all of the since R has no zero divisors

Thus R[x] has no zero divisors

Can someone please confirm that this solution is OK?

Peter - Dec 25th 2012, 08:18 PMDevenoRe: Polynomial rings - zero divisors in R and r[x]
well, no.

c_{k}is a sum of elements, so there is no reason to suppose it can't be 0, even if all the terms in the sum are non-zero.

however, you're close...

c_{m+n}= a_{m}b_{n}≠ 0, since a_{m}≠ 0, and b_{n}≠ 0.

since this is the leading coefficient of f(x)g(x), f(x)g(x) cannot be the 0-polynomial.