Exercise 23, § 1

(a) Let M, N be two subsets of an A-module E and m and n their annihilators; show that the annihilator of $M \cap N$ contains $m + n$ and give an example where it is distinct from $m + n$. (b) In a product module $\prod_t E_t$, the annihilator of a subset F is the intersection of the annihilators of its projections. (c) In a free A-module, the annihilator of an element $\neq 0$ contains only left divisors of 0 in A; in particular, if A is a ring with no divisors of 0, every element $\neq 0$ of a free module is free.