Simple Module - Examples

Examples

Z-modules are the same as abelian groups, so a simple Z-module is an abelian group which has no non-zero proper subgroups. These are the cyclic groups of prime order.

If I is a right ideal of R, then I is simple as a right module if and only if I is a minimal non-zero right ideal: If M is a non-zero proper submodule of I, then it is also a right ideal, so I is not minimal. Conversely, if I is not minimal, then there is a non-zero right ideal J properly contained in I. J is a right submodule of I, so I is not simple.

If I is a right ideal of R, then R/I is simple if and only if I is a maximal right ideal: If M is a non-zero proper submodule of R/I, then the preimage of M under the quotient map R → R/I is a right ideal which is not equal to R and which properly contains I. Therefore I is not maximal. Conversely, if I is not maximal, then there is a right ideal J properly containing I. The quotient map R/I → R/J has a non-zero kernel which is not equal to R/I, and therefore R/I is not simple.

Every simple R-module is isomorphic to a quotient R/m where m is a maximal right ideal of R. By the above paragraph, any quotient R/m is a simple module. Conversely, suppose that M is a simple R-module. Then, for any non-zero element x of M, the cyclic submodule xR must equal M. Fix such an x. The statement that xR = M is equivalent to the surjectivity of the homomorphism R → M that sends r to xr. The kernel of this homomorphism is a right ideal I of R, and a standard theorem states that M is isomorphic to R/I. By the above paragraph, we find that I is a maximal right ideal. Therefore M is isomorphic to a quotient of R by a maximal right ideal.

If k is a field and G is a group, then a group representation of G is a left module over the group ring k. The simple k modules are also known as irreducible representations. A major aim of representation theory is to understand the irreducible representations of groups.