existence of a maximal ideal for a strict ideal

Let $R$ be a Ring and $\mathfrak{a}⊊R$ a strict ideal. Then there exists a maximal ideal $\mathfrak{m}\supseteq \mathfrak{a}$.

Let $M\text{coloneqq}\{\mathfrak{m}\subseteq R|\mathfrak{a}\subseteq \mathfrak{m}⊊R,\mathfrak{m}\text{ ideal}\}$ Then $M$ is nonempty, since $\mathfrak{a}\in M$. Let ${\mathfrak{m}}_1\subseteq {\mathfrak{m}}_2...$ be a chain, then $\bigcup _{i\in \mathbb{N}}{\mathfrak{m}}_i$ is an upper bound (see: Vereinigung von einer Kette von Idealen als Ideal) and also strict, since $1\notin {\mathfrak{m}}_i$ for $i\in \mathbb{N}$ Therefore, we can apply Zorn's lemma and conclude, that a maximal ideal $\mathfrak{m}$ exists.