Example of a maximal ideal

Question

Let $A$ be the set of bounded continuous functions from the set of real numbers to itself. Then $A$ is a ring under pointwise addition and multiplication. The set $I$ of all functions $f \in A$ satisfying $f(x) \to 0$ as $|x| \to \infty$ is an ideal in $A$. Hence by Zorn's lemma there is a maximal proper ideal in $A$ that contains $I$. Can someone give me an "explicit example" of such an ideal?