Exercise 14, § 8

Let $A$ be a commutative ring in which there is only a finite number $n$ of divisors of zero. Show that $A$ has at most $(n+1)^2$ elements. (Let $a_1, \ldots, a_n$ be the divisors of zero. Show that the annihilator $\mathfrak{J}_1$ of $a_1$ has at most $n+1$ elements and that the quotient ring $A/\mathfrak{J}_1$ has at most $n+1$ elements.)