# 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

This contradicts the definition of

_{R}R*∼*=*nV*where*V*is theunique simple left*R*-module, and*n*=*l*(*).We claim that*_{R}R*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*. Since*_{R}R^{k}→ M*can be generated by*_{R}R^{k}*k*elements,*μ*(*M*)*≤ k*. If*M*can begenerated by*k −*1 elements, then there exists an epimorphism_{R}R^{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.
## Comments

## Leave a comment