I am a little bit confused with the definition of finitely presented modules. In Lang's Algebra he defines a module $M$ to be finitely presented if and only if there is a exact sequence $F'\to F\to M \to 0$ such that both $F', F$ are free. However the standard definition I have seen elsewhere only demands $F'$ be finitely generated. Are these two definitions equivalent?
Looking at the situation of a non-principal ideal of a ring, say $(x, y)$ of $\mathbb{R}[x, y]$, it appears that this is finitely presented, by the usual definition, but I don't see any way of making it finitely presented by Lang's definition.
Best Answer
In Lang's Algebra he defines a module $M$ to be finitely presented if and only if there is an exact sequence $F'\to F\to M \to 0$ such that both $F', F$ are free of finite rank, and this is the definition of finitely presented modules. (Note that for each module $M$ there is an exact sequence $F'\to F\to M\to 0$ with $F,F'$ free modules.)
"However the standard definition I have seen elsewhere only demands $F'$ be finitely generated." This is the definition of finitely related modules.
"Are these two definitions equivalent?" In general they aren't: Let $M$ be a finitely presented module, and $L$ a free module which is not finitely generated. Then $M\oplus L$ is finitely related, but not finitely presented. However, if the module is finitely generated the two definitions coincide.