Direct limit of n-presented modules?

Question

A module $M$ is said to be $n$-presented If there exist an exact sequence $$F_{n}\to F_{n-1}\to \cdots \to F_{1}\to F_{0}\to M$$ with each $F_{i}$ is free finitely generated. For example $M$ is $0$-presented if and only if it is finitely generated and $M$ is $1$-presented if and only if it is finitely presented. I know that every module is isomorphic to a direct limit of finitely presented modules. Is it true for any integer $n$? i.e, can say that every module is isomorphic to a direct limit of $n$-presented modules?

Answer 1

No. For example, let $k$ be a field, and let $R$ be the ring $k\oplus V$, where $V$ is an infinite dimensional square zero ideal.

Then $R$ has no $2$-presented modules except the finitely generated projectives, so any direct limit of $2$-presented modules must be a direct limit of projective modules, and therefore flat. But $R$ has modules that are not flat, such as $R/V$.