A (left) artinian ring is a ring for which every descending chain of its (left) ideals stabilizes, i.e. there is such that for all . A ring is artinian if it is both left artinian and right artinian.
In an artinian ring the Jacobson radical 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 is left artinian iff is semisimple in and the Jacobson radical is nilpotent. Artinian rings are intuitively much smaller than generic noetherian rings.