This is related to my question here. My question is as follows. How do I see that a nonnegatively graded algebra $A$ is finitely generated as a $k$-algebra if and only if $A_0$ is finitely generated as a $k$-algebra and $A_{>0}$ is finitely generated as an $A$-module (i.e. as a left ideal of $A$)?

Our algebras here are associative, unital, and not necessarily commutative.