Artinian Ring

An Artinian ring is a ring that is both left and right Artinian. A ring is left Artinian if every descending sequence of left ideals eventually becomes constant, and right Artinian if the corresponding condition holds for right ideals. These are the respective descending chain conditions (Wisbauer 1991, p. 26).

For a field and a positive integer , the quotient ring is Artinian. Its vector space dimension over is , so a strictly descending sequence of ideals must strictly decrease dimension and therefore stop. More generally, every finite-dimensional unital associative algebra over a field is left and right Artinian. A commutative ring with multiplicative identity is Artinian iff it has finite module length as a module over itself. In particular, it is then also a Noetherian ring (Stacks Project n.d., Tag 00J4).