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 _{R}R *∼*= *nV *where *V *is theunique simple left *R*-module, and *n *= *l*(_{R}R).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 _{R}R^{k} → M. Since _{R}R^{k}can be generated by *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.

This contradicts the definition of

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