# Answer to Question #17096 in Abstract Algebra for Hym@n B@ss

Question #17096

Let V be a left R-module with elements e1, e2, . . . such that, for any n, there exists r ∈ R such ren, ren+1, . . . are almost all 0, but not all 0. Show that S := V ⊕ V ⊕• • • is not a direct summand of P := V × V ×• • •

Expert's answer

Assume that

Write

(

where (

Then the exists an index

Let

Then

*P*=*S**⊕**T*for some*R*-submodule*T**⊆**P*.Write

(

*e*1*, e*2*, . . .*)= (*s*1*, s*2*, . . .*) + (*t*1*, t*2*, . . .*)where (

*s*1*, s*2*, . ..*)*∈**S*, and (*t*1*, t*2*, . . .*)*∈**T*.Then the exists an index

*n*suchthat*ti*=*ei*for all*i ≥ n*.Let

*r**∈**R*be such that*re*_{n}, re_{n}_{+1}*,. . .*are almost all 0 but not all 0.Then

*r*(*t*1*, . . . , t*_{n}, t_{n}_{+1}*,. . .*) = (*rt*1*, . . . , rt*_{n−}_{1}*, re*_{n},re_{n}_{+1}*, . . .*)*is nonzero*lies in*S*as well as in*T*,a contradiction.Need a fast expert's response?

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