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If $M$ is finitely generated left module over a left noetherian ring $R$, then $M$ is a noetherian module.
I need to show that an arbitrary ascending chain of submodules of $M$ must satisfy ACC, or equivalently, every submodule of $M$ is finitely generated.
Since $R$ is a left noetherian ring, every left ideal in $R$ is finitely generated. I am trying to find a 1-1 correspondence between left ideals of $R$ and submodules of $M$. Is it a correct way, or how can I show the assertion by a different way? Thanks.