Finitely generated modules over a left Noetherian ring are Noetherian

Given: The hypotheses and objects in the Statement.

A unital ring $R$ is left Noetherian when its left regular module ${}_{R}R$ is Noetherian, and right Noetherian when the right regular module $R_R$ is Noetherian. Unqualified “Noetherian ring” means left Noetherian here; the side is stated whenever both notions occur. (Left and right Noetherian rings).

Let $M$ be a left $R$-module and $S\subseteq M$. The submodule generated by $S$ is $⟨S{⟩}_R:=\bigcap \{N\le M:S\subseteq N\}\mathrm{.}$ The family is nonempty because $M\le M$, and its intersection is a submodule by lem-submodule-criterion-sums-and-intersections. Thus $⟨S{⟩}_R$ is the smallest submodule of $M$ containing $S$, just as def-generated-subgroup defines a generated subgroup. (Generated submodule, cyclic and finitely generated modules, module basis and free module).

A finite direct sum is Noetherian if and only if every summand is Noetherian, and it is Artinian if and only if every summand is Artinian. The empty direct sum is included. (Finite direct sums preserve and reflect Noetherian and Artinian conditions).

For every left $R$-module $M$, the free module $R^{(M)}$ on its underlying set admits a canonical surjection ${\epsilon }_M:R^{(M)}\to M$, determined by ${\epsilon }_M(e_m)=m$. Consequently $M\cong R^{(M)}\mathrm{/}\ker {\epsilon }_M$. (Every module is a quotient of a free module).

In a short exact sequence $0\to N\to M\to Q\to 0$, the module $M$ is Noetherian if and only if $N$ and $Q$ are Noetherian. (Noetherian and Artinian conditions are each exact in short exact sequences).