Question

Show that μ is subadditive, in the sense that μ(M ⊕ N) ≤ μ(M) + μ(N) for finitely generated R-modules M,N.

Answer 1

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.