Module problems and examples
Last update: 31 January 2012
Let be a PID. Let be a finitely generated module. Show that is torsion free if and only if is free.
and this is decomposition given by the decomposition given by the decomposition theorem for finitely generated modules.
Then is a module and hence, it is also a module. Thus is a vector space. There are two natural bases of as a vector space
and since is isomorphic to
we also have natural bases
The matrix of the action of with respect to each of these different bases is
Then let and use to define an action of on . Then
and the are monic polynomials with coefficients in .
- Let be an integral domain and let be an module. Show that is a submodule of .
- Give an example of a commutative ring and an module such that
is not an module.
- Let be an integral domain and let be an module. Show that
is a torsion free module.
- Let be a PID and let be a cuclic module. Show that there exists
- Let be a PID and let . Let
be a factorization of into a product of irreducible elements such that is not associate to any , . Show that
- Let be a PID. Let . Show that is a torsion module if and that is torsion free if
- Let be a finitely generated module over a PID. Show that is the direct sum of and a free module.
- Let be a PID and let be a finitely generated torsion free module. Show that is free.
Notes and References
Where are these from?