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