Maximal ideal contains a zero divisor 

Suppose $R$ is a commutative and unital ring. Let the ideal $I$ be maximal and $a,b$ be (nonzero) zero divisors in $R$.
Show that $ab = 0$ implies $a \in I$ or $b\in I$


We've only had a bit of exposure to ideals: we know that $I$ maximal $\to R/I$ field, a little about the Euclidean algorithm, and the definition of a PID.
I'm not sure how to approach this.  The problem seems simple and I'm probably just missing something.
Should I try assuming $a,b \notin I$ and try to derive a contradiction?

In a commutative ring $R$, a maximal ideal $A$ is prime.
$A$ maximal $\Rightarrow R/A$ is a field $\Rightarrow R/A$ is an integral domain $\Rightarrow A$ is prime.
