Proposition 5

a) Let A be a left Artinian (resp. left Noetherian) ring, and let $\varphi : A\rightarrow B$ be a ring homomorphism that makes B into a finitely generated left A-module. The ring B is left Artinian (resp. left Noetherian).

b) Let A be a left Artinian (resp. left Noetherian) ring, and let $\mathfrak{a}$ be a two-sided ideal of A; the ring $A/\mathfrak{a}$ is left Artinian (resp. left Noetherian).

c) Let $(A_i)_{i\in I}$ be a family of left Artinian (resp. left Noetherian) rings. The ring $\prod_{i\in I}A_i$ is left Artinian (resp. left Noetherian).

We will treat the case of Artinian rings; the case of Noetherian rings is analogous.

Let us prove a). By Proposition 4, the ring $B_s$ is an Artinian left A-module and a fortiori an Artinian left B-module.

Assertion b) follows from assertion a) applied to the canonical homomorphism from A to $A/\mathfrak{a}$.

Let us prove c). Set $A =\prod_{i\in I}A_i$. By assumption, $(A_i)_s$ is an Artinian left $A_i$-module and a fortiori an Artinian left A-module. By Example 5 of VIII, p. 4, the A-module $A_s$ is Artinian.