Let be a nonzero right -module. Then the following are equivalent:
(i) is simple.
(ii) for all
(iii) for some maximal right ideal of .
Proof: (i)=>(ii) Let . is a submodule of . (Let and , then , ). Since is simple, implies .
(ii)=>(i) Condition (ii) implies that is the only nonzero submodule of , thus is simple.
(ii)=>(iii) Let , . is an -linear map that is surjective, thus . is a right ideal of . Since (ii) implies (i), is a simple module. Thus by Correspondence Theorem, is a maximal right ideal.
(iii)=>(i) Follows from the Correspondence Theorem: The map is a bijection from the set of submodules of containing and the submodules of . Thus if is maximal, the only submodules containing are and , thus the only submodules of are and , i.e. is simple.