Finitely generated vs presented

I am curious exactly what are the differences between finitely generated and finitely presented? I understand that finitely generated means we have, for an $R$-module $M$ that there exists an epimorphism $$p:R^n\to M$$ and definitionally that finitely presented is when the kernel of $p$ is finitely generated that is $$h:R^m\to\ker p$$ is an epimorphism, so we get $$R^m\xrightarrow{h} R^n \xrightarrow{p} M\to 0$$ being exact.

What I don't get is what additional information does it supply? Wouldn't the kernel of any such epimorphism be finitely generated? If not got a good example of it not being the case?

Solution 1:

For noetherian rings these properties are indeed equal, but not in general. Take your favorite non-noetherian ring $R$. It has an ideal $I$ which is not finitely generated. Then $R/I$ is finitely generated but not finitely presented.

Added later

Since it is not quite obvious, let me add a reason why $R/I$ being finitely presented would imply $I$ being finitely generated:

If $R/I$ is finitely presented, there is an exact sequence $0 \to K \to R^n \to R/I \to 0$ for some integer $n$ and some finitely generated module $K$. By applying Schanuel's lemma to that sequence and to the canonical exact sequence $0 \to I \to R \to R/I \to 0$, we obtain an isomorphism $R \oplus K \cong R^n \oplus I$. Now we see that $I$ is a quotient of a finitely generated module, and therefore is finitely generated as well.