Finitely generated module question

Question

I'm revising Commutative Algebra and have a quick (possibly stupid) question - I can't get my head around the fact that a submodule of a finitely generated module is not necessarily finitely generated - so M is finitely generated if there is a finite set Y such that very element of M is a linear combination of elements of Y, but if A is a submodule of M any element in A is also an element of M and thus can be written as a linear combination of elements of Y as well?

Any help greatly appreciated, thanks!!

Answer 1

You "proof" doesn't work since $Y$ is not a subset of $A$ (which is the minimal requirement for a generating system).

Rings over which (left) submodules of f.g. (left) modules are f.g. are precisely the (left) noetherian rings.

Answer 2

The problem is that the generators for $M$ may not lie in the submodule $A$. As an example, every ideal $I$ in a commutative ring $R$ is an $R$-submodule of the $R$-module $R$, but there are rings with ideals that are not finitely generated (i.e., non-Noetherian rings), although $R$ is always finitely generated as an $R$-module (by the element $1$). Explicitly, the ideal $(x_1, x_2, ...)$ in the polynomial ring $k[x_1, x_2, ...]$ in infinitely many variables requires infinitely many generators (note: the generator $1$ for $R$ is not contained in $(x_1, x_2, ...)$).