When can the rank of a submodule be bigger than the rank of the module itself?

Question

It is well known that the dimension of a subspace is less than or equal to the dimension of the vector space it is contained in. The same is true e.g. for modules over a principal ring.

I am looking for situations in which what would be naturally called the dimension does not satisfy this property, that is to say a class X of objects for which what would naturally play the role of the dimension does not satisfy this property. Namely, there exists an object $x$ in $X$ and $y$ a subobject of $x$ which is such that $dim(y) > dim(x)$.

The only example I found for the moment is free groups. The "dimension" is then the rank and it is known that given a free group of rank 2, there are subgroups that are isomorphic to free groups of any rank.

Do you have any other example of such a situation? (e.g. modules over some weird ring).

Answer 1

Look at the answer to this StackExchange question: A free submodule of a free module having greater rank the submodule (and also the MO discussion referenced therein).

Answer 2

This is kind of a silly example: $\mathbb{Z} \subseteq \mathbb{Q}$, but the Krull dimension of $\mathbb{Z}$ is one while the Krull dimension of $\mathbb{Q}$ is 0.