Let be a ring. Let be a family of ideals of indexed by a linearly ordered set such that if then . Prove that is an ideal of .
In a previous exercise, we saw that is an additive subgroup. To show that is an ideal, it suffices to show that absorbs on both sides. To that end, let and . Since , we have for some . Since is an ideal, . Thus , and absorbs .
Thus is an ideal.