**Problem**. (McCoy) Let be a commutative ring with identity and let

1) Prove that is a zero-divisor if and only if there exists some such that

2) Prove that if is reduced and for some then for all and

**Solution**. 1) If there exists such that then clearly is a zero-divisor of For the converse, let be a polynomial with minimum degree such that I will show that So, suppose to the contrary, that If for all then for all and so contradicting the minimality of because . So we may assume that the set is non-empty and so we can let

Then

Thus

and so

Hence But we have which is impossible because was supposed to be a polynomial with minimum degree satisfying

2) The proof of this part is by induction over It is obvious from that Now let and suppose that whenever We need to show that whenever So suppose that The coefficient of in is clearly

which after multiplying both sides by gives us

Call this (1). Now in the first sum in (1), since we have and hence by the induction hypothesis Thus So the first sum in (1) is In the second sum in (1), since and we have Therefore by the induction hypothesis and hence So the second sum in (1) is also Thus (1) becomes and so, since is reduced,