Exercise 6, § 8

Let A be a commutative ring and M a submonoid of the additive group of A. Let $n$ be an integer $> 0$ such that $m^n = 0$ for $m \in M$.

(a) If $n!$ is not a divisor of zero in A, the product of a family of $n$ elements of M is zero.

(b) Show that the conclusion in (a) is not necessarily true if $n!$ is a divisor of zero in A.