Problem 1: Let 0 M' MM" →0 be a short exact sequence of R-modules. Prove that: 1. If M' and M" are finitely generated, then so is M. 2. If M is divisible, then so is M". Problem 2: 1. Give an example of an injective module which is not free and another example of a free module which is not injective. 2. Every R-module is projective if and only if every R-module is injective.