The University of British Columbia Sessional Examinations
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چکیده
1. [10 points] Determine whether the following statements are true or false (you have to include proofs/counterexamples): (a) Let R be an integral domain, F – a free R-module of finite rank, and M – a torsion R-module. Then there is no injective homomorphism from F to M . Solution: True. Suppose there was an injective homomorpism φ : F → M . Then let N = φ(F ); N is a submodule of M , and therefore is also a torsion module. On the other hand, N is isomorphic to F (since φ is invective, it is an isomorphism from F onto N), hence free. We proved in class that a free module over an integral domain has to be torsion-free. Hence, Tor(N) = 0; but we also have Tor(N) = N , so N = 0. Thus, φ is a zero homomorphism, and therefore not injective – a contradiction (we are assuming F 6= {0}, i.e. the set of generators of F is non-empty). (b) Over an arbitrary integral domain, any submodule of a free module is free. Solution: This is false. For example, take a non-principal ideal, e.g. (2, x) in Z[x]. It is a submodule of the free Z[x]-module M = Z[x], but it is not free.
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