simple R-module

[Guest rainpurple]

For some hint about the following problem, thx a lot~

 

Let R be a ring with 1. A nonzero left R -module S is simple if 0 and S are the only submodules of S . Let

0 ---> S---(alpha)--->M--(pi)-->S---->0

be a short exact sequence of R -modules which is not split, and such that S is a simple R -module. Show that the only nonzero submodules of M are alpha(S) and M

 

Rp~

[matt grime]

suppose that T were a submodule of M. pi(T) is a submodule of S, hence it is 0 or S, which implies T is either alpha(S) or M.