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 *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 a simple module is a division ring.

*Need examples, connection to semisimple modules*