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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.
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.
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.
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