It is common knowledge that submodules of finitely generated modules need not be finitely generated. One can see this by considering a non-Notherian ring and extending this as a module. My question is, how does one think about this intuitively? The way that I think about is:
When I restrict myself to a subspace, there is a chance I no longer have access to the elements that finitely generate the whole module. In contrast with Linear Algebra, this can occur when the subspace is non-empty.
Could someone let me know if this is a correct way to view this or if there is a more intuitive way of viewing this result?