Corollary

The ring A is left Artinian (resp. left Noetherian) if and only if the ring B is.

Because of the isomorphism between the ordered set of left ideals of B and the set of A-submodules of P, the ring B is left Artinian (resp. left Noetherian) if and only if the A-module P is Artinian (resp. Noetherian). However, by Theorem 1 of VIII, p. 101, the A-module P is generating and finitely generated; in particular, $A_s$ is isomorphic to a direct factor of $P^n$ for an integer $n\geqslant 1$. Consequently, P is Artinian (resp. Noetherian) if and only if A is left Artinian (resp. left Noetherian).