# Answer to Question #17188 in Algebra for Hym@n B@ss

Question #17188

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

Expert's answer

Let

*{x*1*, . . . , xn}*begenerators for*M*, and*{y*1*, . . . , ym}*be generators for*M*.Then*x*1 ,*y*1*, . . . , xn*,*ym*generate*M*=*M*1*⊕**M*2, or maybe some of them can be excluded. Then*μ*(*M**⊕**N*)*≤**μ*(*M*) +*μ*(*N*) as desired.
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