Answer to Question #17096 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 S := V ⊕ V ⊕• • • is not a direct summand of P := V × V ו • •

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 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(t1, . . . , t_n, t_n+1,. . .) = (rt1, . . . , rt_n−1, re_n,re_n+1, . . . )
is nonzero lies in S as well as in T,a contradiction.