Artinian ring

Remarks

Example: $\Z/n\Z$ for $n \ge 1$ is Artinian, because it has only finitely many ideals, so a decreasing chain takes only finitely many values. Example: $k[x]/(x^2)$ over a field $k$ is Artinian, because every ideal is a $k$-subspace of a $2$-dimensional space. Non-example: $\Z$ is not Artinian, because $(2) \supsetneq (4) \supsetneq (8) \supsetneq \cdots$ never stops. Equivalently, $R$ is Artinian when it is an Artinian module over itself, or when every nonempty set of ideals has a minimal element. For a noncommutative ring one defines left Artinian and right Artinian rings with left and right ideals.