Exercise 10, § 1

Let A be a ring and $\mathfrak{a}$ and $\mathfrak{b}$ two-sided ideals of A. If the rings $A/\mathfrak{a}$ and $A/\mathfrak{b}$ are left Artinian (resp. left Noetherian), then so is the ring $A/\mathfrak{a}\cap \mathfrak{b}$; by contrast, the ring $A/\mathfrak{a}\mathfrak{b}$ is not necessarily left Artinian (resp. left Noetherian).