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_{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.