Answer to Question #17186 in Algebra for Hym@n B@ss

Question

For any finitely generated left module M over a ring R, let μ(M) denote the smallest number of elements that can be used to generate M. If R is an artinian simple ring, find a formula for μ(M) in terms of l(M), the composition length of M.

Answer 1

Say _RR ∼= nV where V is theunique simple left R-module, and n = l(_RR).We claim that μ(M)= [l(M)/n], [α] is defined to be the smallest integer ≥ α. To prove this formula, let l =l(M), and k = [l/n]. Since l ≤ kn, thereexists an epimorphism _RR^k → M. Since _RR^kcan be generated by k elements, μ(M) ≤ k. If M can begenerated by k − 1 elements, then there exists an epimorphism _RR^k−1→ M, and we get l(M) ≤ l(R^k−1)= (k − 1)n.
This contradicts the definition of k,so we must have μ(M)= k, as claimed.