Some more properties of ideal arithmetic

Let R be a ring and let I,J,K \subseteq R be ideals.

  1. Prove that I(J+K) = IJ+IK and (I+J)K = IK+JK.
  2. Prove that if J \subseteq I, then (J+K) \cap I = J+(K \cap I).

  1. We show that I(J+K) = IJ + IK; the proof of the other equality is similar. (\subseteq) Let \alpha \in I(J+K). Then \alpha = \sum a_i(b_i+c_i) for some a_i \in I, b_i \in J, and c_i \in K. Then \alpha = \sum (a_ib_i + a_ic_i) = (\sum a_ib_i) + (\sum a_ic_i) \in IJ + IK. (\supseteq) Note that since J \subseteq J+K, IJ \subseteq I(J+K). Similarly, since K \subseteq J+K, IK \subseteq I(J+K). By this previous exercise, IJ+IK \subseteq I(J+K).
  2. (\subseteq) Let x \in (J+K) \cap I. Then x \in I and x = y+z for some y \in J and z \in K. Since J \subseteq I, x-y = z \in I. Thus z \in K \cap I, and x = y+z \in J+(K \cap I). (\supseteq) Let x \in J+(K \cap I). Then x = y+z where y \in J and z \in K \cap I. Again because J \subseteq I, we have x = y+z \in I. Moreover, x = y+z \in J+K. Thus x \in (J+K) \cap I.
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