Let $A$ be a nonnegatively graded algebra and $M$ a nonnegatively graded $A$-module. Then, $A_{>0}M$ is a graded $A$-submodule of $M$.

My question is as follows. Does it follow that $M$ is finitely generated as an $A$-module if and only if $M/A_{>0}M$ is finitely generated as an $A$-module?

Is this well-known? Can I find a proof of this anywhere? Or could anybody supply a proof?

Let $A$ be a nonnegatively graded algebra and $M$ a nonnegatively graded $A$-module. Then, $A_{>0}M$ is a graded $A$-submodule of $M$.

(Here, given a nonnegatively graded algebra $A$, we've defined $A_{>0} := \oplus_{i > 0} A_i$.)

My question is as follows. How do I see that $M$ is finitely generated as an $A$-module if and only if $M/A_{>0}M$ is finitely generated as an $A$-module?

Is this well-known? Can I find a proof of this anywhere? Or could anybody supply a proof?