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.

- artinian object?

Revised on December 6, 2011 21:23:57
by Tim Porter
(95.147.237.250)