Simple module

In abstract algebra, a (left or right) module S over a ring R is called simple if it is not the zero module and if its only submodules are 0 and S. Understanding the simple modules over a ring is usually helpful because they form the "building blocks" of all other modules in a certain sense.

Examples

Abelian groups are the same as Z-modules. The simple Z-modules are precisely the cyclic groups of prime order.

If K is a field and G is a group, then a representation of G is nothing but 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 list those irreducible representations for a given group.

Facts

The simple modules are precisely the modules of length 1; this is nothing but a reformulation of the definition.

Every simple module is indecomposable, but the converse is in general not true.

Not every module has a simple submodule; consider for instance the Z-module Z in light of the second example above.

If S is a simple module and f : S → T is a module homomorphism, then f is either zero or injective. (Reason: the kernel of f is a submodule of S and hence is either 0 or S.) If T is also simple, then f is either zero or an isomorphism. (Reason: the image of f is a submodule of T and hence either 0 or T.) Taken together, this implies that the endomorphism ring of any simple module is a division ring. This result is known as Schur's lemma.

The converse of Schur's lemma is not true in general: there are non-simple modules whose endomorphism ring is a division ring.

See also

• semisimple modules are modules that can be written as a sum of simple submodules
• simple groups are similarly defined to simple modules