Corollary

Under the hypotheses of Proposition 8, if $B$ is left Noetherian (resp. Artinian), so is $A$.

If (a) is a non-stationary increasing (resp. decreasing) sequence of left ideals of $A$, the sequence $(\mathrm{Ba}_n)$ of ideals of $B$ is non-stationary and increasing (resp. decreasing) since $\rho^{-1}(\mathrm{Ba}_n) = a_n$ contrary to the hypothesis.

Remark (1). If $A$ and $B$ are commutative, we shall see in Chapter II, § 2, no. 5, Corollary 4 to Proposition 11, that the hypothesis of Proposition 8 implies that for every prime ideal $p$ of $A$, there exists a prime ideal $q$ of $B$ such that $\rho^{-1}(q) = p$ (where $p = A \cap q$ when $A$ is identified with a subring of $B$).

An important application of Proposition 8 is when $B$ is itself a faithfully flat $A$-module. But in this case we have the following more precise proposition: