Let be a commutative ring with . Let be an element of the polynomial ring . Prove that is a zero divisor in if and only if there is a nonzero such that .
If for some nonzero , then letting , is a zero divisor.
Now suppose is a zero divisor; that is, for some , . We may choose to have minimal degree among the nonzero polynomials with this property.
We will now show by induction that for all .
For the base case, note that . The coefficient of in this product is on one hand, and 0 on the other. Thus . Now , and the coefficient of in is . Thus the degree of is strictly less than that of ; since has minimal degree among the nonzero polynomials which multiply to 0, in fact . More specifically, for all .
For the inductive step, suppose that for some , we have for all . Now . On one hand, the coefficient of is , and on the other hand, it is 0. Thus . By the induction hypothesis, if , then . Thus all terms such that are zero. If , then we must have , a contradiction. Thus we have . As in the base case, and has degree strictly less than that of , so that by minimality, .
By induction, for all . In particular, . Thus .