M a finitely generated module over a commutative ring A. I can't think of an example of two maximal linearly independent subsets of M having different cardinality. I know that they all have the same cardinality if A is integral domain. Any suggestions are welcome!



Cardinality of maximal linearly independent subset
I found an old paper by Lazarus (Les familles libres maximales d'un module ont-elles le meme cardinal?, Pub. Sem. Math. Rennes 4 (1973), 1-12) which contains the the following result: Let A be a commutative ring with unit and M an A-module. In the following situations, maximal linearly independent subsets of M have the same cardinality:


1. If M is a free A-module of infinite rank.
2. If A is reduced and has only finitely many minimal primes (e.g. integral domain, reduced Noetherian ring)
3. If A is Noetherian and M is a free A-module.
4. If A is Noetherian and M is a submodule of a free A-module of finite rank.
5. If A is Noetherian and M has an infinite linearly independent subset.
6. If A is Noetherian and M is a submodule of a flat A-module.
7. If A is Artin local and the zero ideal $(0)\subset A$ is irreducible.


And the examples given in the paper of modules not satisfying this same cardinality property are highly nontrivial.


