Show that μ is subadditive, in the sense that μ(M ⊕ N) ≤ μ(M) + μ(N) for finitely generated R-modules M,N.
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Expert's answer
2012-11-19T07:55:21-0500
Let {x1, . . . , xn} begenerators for M, and {y1, . . . , ym} be generators for M.Then x1 , y1, . . . , xn , ym generate M = M1⊕M2, or maybe some of them can be excluded. Then μ(M ⊕ N) ≤ μ(M) + μ(N) as desired.
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