Let $R$ be an associative $\mathbb{C}-$algebra with unity $1$, and for all $i\in\mathbb{N}$, let $R_i\leq R$ finitly generated subalgebras with the same unity, such that $R_i\leq R_{i+1}$ and $R=\cup R_i$.
Take $V$ an $R-$module and $V_i$ $R_i-$modules (finitely generated) , such that $V_i\leq V$ (it's a $R_i$ submodule of $V$, and we view $V$ as a $R_i$ module by restriction). And for all i let $\psi_i:V_i\rightarrow V_{i+1}$ and injective morphism of $R_i-$modules ($V_{i+1}$ as a $R_i$ module by restriction)
Does exits a collection $\{W_i\}_{i\in \mathbb{N}}$ such that
- $W_i$ is a $R_i-$module
- $W_i\leq W_{i+1}\leq V$
- $W_i\simeq V_i$
No. For instance, let $R_i=R=\mathbb{C}[x,y]$ for all $i$, $V=R$, $V_i=R$ if $i$ is even and $V_i=(x,y)\subset R$ if $i$ is odd. We can take $\psi_i$ to be the inclusion if $i$ is odd and multiplication by $x$ if $i$ is even.
Now suppose the submodules $W_i$ which you ask for existed. Note that since $V_i\not\cong V_{i+1}$ for each $i$, $W_i$ would have to be a proper submodule of $W_{i+1}$ for each $i$. But then the $W_i$ are an infinite strictly ascending chain of ideals in $R$, which is impossible as $R$ is Noetherian.