Let R be a commutative ring with identity and let be a polynomial in
Prove that is a zero divisor of if and only if there is a nonzero element such that
Proof so far.
Suppose that is a zero divisor, then by definition such that
But how would I factor this down into one single element?
Conversely, suppose that , then would I be able to find a polynomial that behaves like s?
Thank you!
Thank you!!!
One last question:
Prove that if R has no nonzero nilpotent elements and for some , then .
Proof so far.
Suppose that for some element , if we have , then .
Now, let for some
So the product of each coefficient is also zero, how should I proceed?
proof by induction over it's obvious from that now let and suppose whenever we need to show that whenever
so suppose the coefficient of in is clearly which after mutiplying both sides by gives us:
now in the first sum in (1), since we have hence by induction hypothesis thus i.e. the first sum is equal to in the second sum in (1), since
and we must have thus by induction hypothesis and hence so the second sum is as well. hence (1) becomes: i.e. is
nilpotent. but has no nonzero nilpotent element. thus