nLab
artinian ring

A (left) artinian ring $R$ is a ring for which every descending chain $R=I_0\supset I_1\supset I_2\supset \dots \supset I_n\supset \dots$ of its (left) ideals stabilizes, i.e. there is $n_0$ such that $I_{n+1}=I_n$ for all $n\ge n_0$. A ring is artinian if it is both left artinian and right artinian.

In an artinian ring $R$ the Jacobson radical $J(R)$ is nilpotent. A left artinian ring is semiprimitive if and only if the zero ideal is the unique nilpotent ideal.

A dual condition is noetherian: a noetherian ring is a ring satisfying the ascending chain condition on ideals. The symmetry is severely broken if one considers unital rings: for example every unital artinian ring is noetherian. For a converse there is a strong condition: a left (unital) ring $R$ is left artinian iff $R/J(R)$ is semisimple in ${}_R\mathrm{Mod}$ and the Jacobson radical $J(R)$ is nilpotent. Artinian rings are intuitively much smaller than generic noetherian rings.

Related concept

• artinian object?