Maximal linearly independent sets in a f.g. module

Question

Suppose $M$ is a finitely generated module over a commutative unital ring $R$.

Is it true that every maximal linearly independent set in $M$ has the same size?

What is the most general condition on $R$ for this to be true?

Answer 1

From linear algebra we know this when $R$ is a field. It follows more generally when $R$ is an integral domain, since every maximal linearly independent subset of the $R$-module $M$ induces a maximal linearly independent subset of the $Q(R)$-module $(R \setminus \{0\})^{-1} M$ (this can be checked by a direct calculation, for instance).

There are many more conditions under which it is true. But for the general case there are counterexamples. For details, see the identical question MO/30066 and the paper

M. Lazarus, Les familles libres maximales d'un module ont-elles le meme cardinal?, Pub. Sem. Math. Rennes 4 (1973), 1-12