existence of a maximal ideal for a strict ideal

Let \(R\) be a Ring and \(\mathfrak{a} \subsetneq R\) a strict ideal. Then there exists a maximal ideal \(\mathfrak{m} \supseteq \mathfrak{a}\).

Let \(M \coloneqq \{\mathfrak{m} \subseteq R \vert \mathfrak{a} \subseteq \mathfrak{m} \subsetneq R, \mathfrak{m} \text{ ideal}\}\) 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 \(1 \not\in \mathfrak{m}_i\) for \(i \in \mathbb{N}\) Therefore, we can apply Zorn's lemma and conclude, that a maximal ideal \(\mathfrak{m}\) exists.