A question about a Noetherian ring starting from a subring

Question

Proposition. Let $A$ be a subring of $B$, $A$ be a Noetherian, and $B$ finitely generated as an $A-$ module. Then $B$ is a Noetherian ring.

Proof. Since $B$ is finitely generated as an $A-$ module, then $B$ is Noetherian as an $A-$ module.

Question How can I conclude that $B$ is a Notherian ring? Where do we use the fact that $A$ is a sub-ring of $B$?

Edit. Does this result also apply to Artinian rings?

Answer 1

Every $B$-submodule of $B$ is an $A$-submodule, so is finitely generated over $A$, so is finitely generated over $B$.