Math 121 Homework 5: Notes on Selected Problems
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چکیده
12.1.2. Let M be a module over the integral domain R. (a) Assume that M has rank n and that x1, . . . , xn is any maximal set of linearly independent elements of M . Prove that N is isomorphic to Rn and that the quotient M/N is a torsion Rmodule. (b) Prove conversely that if M contains a submodule N that is free of rank n (i.e., N Rn) such that the quotient M/N is a torsion R-module then M has rank n.
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