This is a true in a commutative ring with $1$, but does it also hold in a noncommutative ring with $1$? The proof in my book is just an application of Zorn's lemma, but the commutativity of the ring is not used anywhere.
The theorem is known as the Krull's Theorem and stated in its complete form it says:
Let $R$ be a ring with identity, and let $I$ be a (left, right, two-sided) ideal of $R$ that is distinct from $R$. Then there exists a maximal (left, right, two-sided) ideal of $R$ containing $I$.
The proof is similar to the standard proof given when $R$ is commutative.