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

Question

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 ו • • is not a semisimple R-module.

Answer 1

Assume that P = S ⊕ T for some R-submodule T ⊆ P. Write (e1, e2, . . .) =(s1, s2, . . .) + (t1, t2, . . . )where (s1, s2, . . . ) ∈ S, and (t1, t2, . . . ) ∈ 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(t1, . . . , t_n,t_n+1, . . .) = (rt1, . . . , rt_n−1,re_n, re_n+1, . . . ) <>0 lies in S as well as in T, a contradiction.