Let $R$ be a ring. Recall that a module $M$ is called finitely presented if there is an exact sequence


$R^n \to R^m \to M \to 0$.


with $n,m \in \mathbb{N}$.
A well known result states that any module $M$ is a filtered direct limit of finitely presented modules. A bit more generally, we can call a module $M$ 2-finitely presented if there is an exact sequence


$R^k \to R^n \to R^m \to M \to 0$


with $k,m,n \in \mathbb{N}$.


My question is: is any module the filtered direct limit of 2-finitely presented modules? what about higher order presentation?



Direct limit and finite presentation of modules
Assume that a f.p. module $M$ is filtered direct limit of modules $(M\_i)$.


Recall that this $M$ f.p. means that $\mathrm{Hom}(M,--)$ commutes with inductive limits. So the identity of $M$ can be lifted to $M\to M\_i$ for $i$ large enough (say $i\ge i\_0$), let $j\_i$ be the lift $M\to M\_i$, and let $N\_i$ be the kernel of $M\_i\to M$. Thus for $i\ge i\_0$ we have $M\_i=j\_i(M)\times N\_i$. But a direct factor of a module with your additional condition (I think it's called $FP\_3$-Property) is still $FP\_3$, so it's impossible if $M$ itself is not $FP\_3$. So your expectation is hopeless: there is no way to "approximate" by modules with higher finiteness properties using direct limits. 


