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.

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