artinian ring

A (left) artinian ring RR is a ring for which every descending chain R=I 0I 1I 2I nR=I_0\supset I_1\supset I_2\supset \ldots \supset I_n\supset\ldots of its (left) ideals stabilizes, i.e. there is n 0n_0 such that I n+1=I nI_{n+1}=I_n for all nn 0n\geq n_0. A ring is artinian if it is both left artinian and right artinian.

In an artinian ring RR the Jacobson radical J(R)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 RR is left artinian iff R/J(R)R/J(R) is semisimple in RMod_R Mod and the Jacobson radical J(R)J(R) is nilpotent. Artinian rings are intuitively much smaller than generic noetherian rings.