# Answer to Question #17097 in Abstract Algebra for Hym@n B@ss

Question #17097

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 P := V × V ×• • • is not a semisimple R-module.

Expert's answer

Assume that

*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*such that*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}*, . . .*)*<>*0 lies in*S*as well as in*T*, a contradiction.
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