Corollary

Every left Artinian ring is left Noetherian.

Let A be a left Artinian ring. By Theorem 1, the A-module $A_s$ has finite length. We then apply Proposition 1 of VIII, p. 2.

Let A be a left (resp. right) Artinian ring; the length of the A-module $A_s$ (resp. $A_d$) (I, §4, No. 7, p. 44) is called the left (resp. right) length of the ring A. When A is a commutative Artinian ring, these two lengths coincide and are simply called the length of A. When A is left and right Artinian but is not commutative, the left and right lengths of A are not necessarily equal (VIII, p. 14, Exercise 3).