Let $R$ be a commutative ring and $M$ an $R$-module. The module $M$ is finitely generated iff there is an exact sequence $R^{k_0} \to M \to 0$. Similarly, $M$ is finitely presented iff there is an exact sequence $R^{k_1} \to R^{k_0} \to M \to 0$. It seems we could generalize this as follows: for $n \in \mathbb{Z}_{\ge 0}$ let us call $M$ a finitely $n$-presented module, if there is an exact sequence $R^{k_n} \to \dotsm \to R^{k_0} \to M \to 0$. So finitely generated = finitely $0$-presented, and finitely presented = finitely $1$-presented. You could also define finitely $\infty$-presented modules, which have a resolution with finite free modules. I would even try defining $M$ to be finitely $\omega$-presented, if it has a finite free resolution $0 \to R^{k_m} \to \dotsm \to R^{k_0} \to M \to 0$ (I think these have been studied a lot).
I haven't seen these notions defined before (except for the last one). Couldn't they be useful, or have they been used? I think both finitely generated and finitely presented modules are important, although if you are only interested in noetherian rings, there is no difference.
For example, there is a result saying that if $0 \to M' \to M \to M'' \to 0$ is exact, $M''$ is finitely presented and $M$ is finitely generated, then $M'$ is also finitely generated. You could generalize this: if $M$ is finitely $n$-presented and $M''$ is finitely $(n+1)$-presented, then $M'$ is finitely $n$-presented.
Also we could look at submodules: $M$ is noetherian iff every submodule of $M$ is a finitely generated. $M$ is coherent iff it is finitely generated, and every finitely generated submodule of $M$ is finitely presented. We could generalize this as follows: $M$ is $n$-coherent ($n \in \mathbb{Z}_{\ge 0}$) iff it is $(n-1)$-presented, and every finitely $(n-1)$-presented submodule of $M$ is finitely $n$-presented. So noetherian = $0$-coherent, and coherent = $1$-coherent. You could also define $R$ to be $n$-coherent iff it is an $n$-coherent $R$-module. The category of noetherian/coherent $R$-modules is abelian, so I guess the same should hold for the category of $n$-coherent $R$-modules.
