In mathematics, specifically in ring theory, the simple modules over a ring R are the (left or right) modules over R which have no nonzero proper submodules. Equivalently, a module M is simple if and only if every cyclic submodule generated by a nonzero element of M equals M. Simple modules form building blocks for the modules of finite length, and they are analogous to the simple groups in group theory.
In this article, all modules will be assumed to be right unital modules over a ring R.
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Examples
Zmodules are the same as abelian groups, so a simple Zmodule is an abelian group which has no nonzero 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 nonzero right ideal: If M is a nonzero 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 nonzero 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 I is a maximal right ideal: If M is a nonzero 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 nonzero kernel which is not equal to R/I, and therefore R/I is not simple.
Every simple Rmodule is isomorphic to a quotient R/m where m is a maximal right ideal of R.^{[1]} By the above paragraph, any quotient R/m is a simple module. Conversely, suppose that M is a simple Rmodule. Then, for any nonzero 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 kG. The simple kG modules are also known as irreducible representations. A major aim of representation theory is to understand the irreducible representations of groups.
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