Why is it so? Herstein's definition of irreducible module is this:

is said to be an irreducible

module if

and if the only submodules of

are

and

When you say "reducible", do you mean not irreducible? Do you use the same definition of "irreducible" as Herstein does? I'm asking because I don't see why what you said is equivalent to

either having a trivial scalar multiplication or having a nontrivial proper submodule.

Of course, if

then

is a nontrivial proper submodule. But I don't understand why the other implication would hold. Say

contains a nontrivial proper submodule

Then it's not true for general modules that

right?