Is a module a limit of n-presented modules?

Question

Let n be an integer. A module M is said to be n-presented if there exist an exact sequence of the form $$ F_{n}\to F_{n-1}\to ...\to F_{1}\to F_{0}\to M \to 0$$ with every $F_{i}$ is a finitely generated free module for example if n=0 M is said to be finitely generated and for n=1 M is called finitely presented. We know that every module is isomorphic to a filtred colimit of finitely presented module. I want to know if the results still true for any n, i.e, Is every module is a direct limit of n-presented modules ?

Answer 1

Edit: the question has now changed. This answer addressed the original question, whether any module is a direct limit of $n$-presented submodules. (The question now asks about a direct limit of $n$-presented modules.)

Let $k$ be a field and let $R=k[x_1, x_2, x_3, \ldots]$. Then $k$ has the structure of an $R$-module with each $x_i$ acting trivially. The module $k$ has no nontrivial submodules and is not itself $n$-presented for any positive integer $n$, so it won't be the direct limit of its $n$-presented submodules.