Simple module
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.
