**Problem**. Let be a commutative ring and suppose that is injective as -module. Prove that for any ideals of we have

**Solution.** The inclusion** ** is trivial. Now let See that the map defined by is a well-defined -homomorphism. Hence, since is self-injective, there exists some such that for all Therefore if we let we will get for all and if we let we will get for all Thus and So

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