Thread: Polynomial rings - zero divisors in R and r[x]

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

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

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_mb_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.