Definition of finitely presented $R$-module

Question

I found in books two different definitions of a finitely presented $R$-module (with $R$ is a commutative ring).

$*$ the $1$st is :

We say $M$ is a finitely presented $R$-module if there exist an exact sequence $$F_0⟶ F_1 ⟶ M ⟶ 0 $$ such that $F_0$ and $F_1$ are free $R$-modules.

$*$ the $2$nd is :

We say $M$ is a finitely presented $R$-module if there exist an exact sequence $$F_0⟶ F_1 ⟶ M ⟶ 0 $$ such that $F_0$ is a finitely generated $R$-module and $F_1$ is a free $R$-module.

Are these two definitions equivalent?

Answer 1

If you replace everywhere ‘free’ by ‘free and finitely generated’ then it becomes correct, and does not require $R$ to be commutative.

A module $M$ is finitely generated precisely when there exists an epimorphism $R^n\twoheadrightarrow M$ for some positive integer $n$. We say that $M$ is finitely presented if moreover the kernel of any such map is again finitely generated.