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

Question #17186
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.
Expert's answer
Say RR &sim;= nV where V is theunique simple left R-module, and n = l(RR).We claim that &mu;(M)= [l(M)/n], [&alpha;] is defined to be the smallest integer &ge; &alpha;. To prove this formula, let l =l(M), and k = [l/n]. Since l &le; kn, thereexists an epimorphism RRk &rarr; M. Since RRkcan be generated by k elements, &mu;(M) &le; k. If M can begenerated by k &minus; 1 elements, then there exists an epimorphism RRk&minus;1&rarr; M, and we get l(M) &le; l(Rk&minus;1)= (k &minus; 1)n.
This contradicts the definition of k,so we must have &mu;(M)= k, as claimed.

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