Let $R$ be a ring. Let $A$ be a commutative subring of $R$. Show that $A$ is contained in a maximal commutative subring of $R$.

I'd like to use Zorn's Lemma. 

I'd like to start by proving that the family $F$ of commutative subrings of R that contain A is inductive. 

In other words, every chain in $F$ (ordered by inclusion) has a maximal element.

But what is the connection between commutativity and this inductive property?