Exercise 16, § 2

Let A be a commutative ring and u = \sum_{k=0}^m a_k X^k a divisor of 0 in the ring A[X]. Show that if v = \sum_{k=0}^n b_k X^k is an element ≠0 of A[X] of degree n such that uv = 0, there exists a polynomial w ≠ 0 of degree ≤ n − 1 such that uw = 0. (Reduce it to the case where b_0 ≠ 0; if a_kv = 0 for 0 ≤ k ≤ m − 1, show that we can take w = b_0; if a_kv = 0 for 0 ≤ k < p ≤ m − 1 and a_p v ≠ 0, show that a_p b_0 = 0 and therefore we can take w = \sum_{k=0}^{n-1} a_p b_{k+1} X^k.) Deduce that there exists an element c ≠ 0 of A such that cu = 0.