Question

If R is a commutative ring or a left noetherian ring, show that any finitely generated artinian left R-module M has finite length.

Answer 1

If R is left noetherian, thenM is a noetherian (as well as artinian) module, so it has finite length.Now assume R is commutative. Since M is a finite sum of cyclicartinian submodules, we may assume M itself is cyclic. Represent M inthe form R/I, where I ⊆ R is a leftideal. Since R is commutative, I is an ideal, and the fact that Mis artinian implies that R/I is an artinian ring. By theHopkins–Levitzki theorem, R/I is also a noetherian ring, so we are backto the case considered above.