## The union of a chain of ideals is an ideal

Let $R$ be a ring. Let $\{I_k\}_{k \in K}$ be a family of ideals of $R$ indexed by a linearly ordered set $K$ such that if $i \leq j$ then $I_i \subseteq I_j$. Prove that $S = \bigcup_{K} I_k$ is an ideal of $R$.

In a previous exercise, we saw that $S \subseteq R$ is an additive subgroup. To show that $S$ is an ideal, it suffices to show that $S$ absorbs $R$ on both sides. To that end, let $r \in R$ and $s \in S$. Since $s \in S$, we have $s \in I_k$ for some $k \in K$. Since $I_k$ is an ideal, $rs, sr \in I_k$. Thus $rs, sr \in S$, and $S$ absorbs $R$.

Thus $\bigcup_{K} I_k \subseteq R$ is an ideal.