"Finitely generated as an $R$-module"

Question

Please could somebody explain to me what it means for something to be finitely generated as an $R$-module? I can't seem to find a definition anywhere! Thanks!

Answer 1

What follows does not literally answer the OP's question (this is done in the other answer), but may help clear up some confusion:

The OP states in comments that they are confused about the expression "finitely generated as an $R$-module". This phrasing is used in contexts when an $R$-module $M$ might also be being considered with various subsidiary structures, e.g. perhaps also as an abelian group, or perhaps as an $S$-module, where $S$ is a subring or an overing of $R$.

In this context, an author might like to emphasize that $M$, when thought of as an $R$-module (rather than with any of its other structures that might be in play), is finitely generated. This is usually expressed via the phrase "$M$ is finitely generated as an $R$-module".

(E.g. if $S$ is a subring of $R$, this is weaker than saying that $M$ is f.g. as an $S$-module, while if $S$ is an overing of $R$, it is stronger than saying that $M$ is f.g. as an $S$-module.)

Mathematically, the meaning is identical to just writing that $M$ is a f.g. $R$-module. Stylistically, it puts more emphasis on the fact that it is the $R$-module structure that is under consideration in the assertion, rather than some other structure.

Answer 2

It means that there is a finite set $\{m_1, \ldots m_k \} \subseteq M$ such that

$$M=\{r_1m_1+\cdots + r_km_k| r_1, \ldots ,r_k \in R\}$$