The set of ideals of a ring is closed under arbitrary intersections

Let R be a ring.

  1. Prove that if I and J are ideals of R, then I \cap J is also an ideal of R.
  2. Prove that if I_k is an ideal of R for each k \in A, A an arbitrary set, then \bigcap_A I_k is also an ideal of R.

  1. In this previous exercise, we showed that I \cap J is a subring of R. Thus it suffices to show that I \cap J absorbs R on the right and the left. To that end, let r \in R and x \in I \cap J. Now x \in I, so that rx, xr \in I. Likewise, rx, xr \in J. Thus rx, xr \in I \cap J, and I \cap J \subseteq R is an ideal.
  2. Again, we showed in this previous exercise that \bigcap_A I_k is a subring of R, os it suffices to show absorption. Let r \in R and x \in \bigcap_A I_k. Now x \in I_k for each k \in A, so that rx, xr \in I_k for all k \in A. Thus xr, rx \in \bigcap_A I_k. Hence \bigcap_A I_k is an ideal of R.
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