A simple module over a ring is a module that is simple as a group with operators—that is, it is a module with no submodules other than itself and the zero module, and it is not itself the zero module and scalar products are not all equal to 0.
If is a commutative ring, then every simple module over is isomorphic (as an -module) to a quotient ring of by a maximal ideal; that is, every simple module over is isomorphic (as an -module) to a quotient ring of that is a field. This is not the case when is not commutative. In this case, every simple left -module is isomorphic (as a left -module) to the quotient of by a maximal left ideal.
For example, all simple modules over the ring of integers are of the form , where is a prime. A more interesting example of a simple module is the (left) module of complex numbers over the ring of complex numbers with a noncommuting indeterminate adjoined, where corresponds to complex conjugation.