57 284
Assignments Done
Successfully Done
In February 2018
Your physics homework can be a real challenge, and the due date can be really close — feel free to use our assistance and get the desired result.
Be sure that math assignments completed by our experts will be error-free and done according to your instructions specified in the submitted order form.
Our experts will gladly share their knowledge and help you with programming homework. Keep up with the world’s newest programming trends.

Answer on Algebra Question 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 ∼= 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 RRk → M. Since RRkcan be generated by k elements, μ(M) ≤ k. If M can begenerated by k − 1 elements, then there exists an epimorphism RRk−1→ M, and we get l(M) ≤ l(Rk−1)= (k − 1)n.
This contradicts the definition of k,so we must have μ(M)= k, as claimed.

Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!


No comments. Be first!

Leave a comment

Ask Your question