The Fundamental Theorem of Comodules (aka the Finiteness Theorem for Comodules) states that if $\Bbbk$ is a field, then any element of a comodule over a $\Bbbk$-coalgebra lies in a finite-dimensional subcomodule. I know that this is a distinctive property of $\Bbbk$ being a field.
Some years ago I run into a paper/book in which the author showed that this is not true anymore already when $\Bbbk$ is a commutative ring, but at the present moment I am not able to find that reference again.
Since I need it for didactic reasons, I would like to ask if anybody is aware of some reference in which the topic is treated with some details and, in particular, where I may find a counterexample when $\Bbbk$ is not a field.