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
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.