Let be a ring with 1. A (left, unital) -module is called irreducible if and 0 and are the only -submodules of . Prove that is irreducible if and only if and if is nonzero, then . Describe the irreducible -modules.
Suppose is irreducible. Now by definition. Now let be nonzero. Note that is a nonzero submodule since ; since is irreducible, . Thus any nonzero element of generates .
Now suppose and that if , then . Let be a nonzero submodule. Now there exists some nonzero , and . Thus . Since the only submodules of are 0 and , is irreducible.
Recall that -modules are precisely the abelian groups, and that -submodules are precisely the subgroups of abelian groups. Now let be an irreducible -module. In particular, is cyclic as an abelian group. So must be finite of prime order, as otherwise it has nontrivial proper subgroups. Conversely, every cyclic group of prime order is irreducible as a -module.