# Simple module

A simple module over a ring $R$ 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 $R$ is a commutative ring, then every simple module over $R$ is isomorphic (as an $R$-module) to a quotient ring of $R$ by a maximal ideal; that is, every simple module over $R$ is isomorphic (as an $R$-module) to a quotient ring of $R$ that is a field. This is not the case when $R$ is not commutative. In this case, every simple left $R$-module is isomorphic (as a left $R$-module) to the quotient of $R$ by a maximal left ideal.

For example, all simple modules over the ring of integers $\mathbb{Z}$ are of the form $\mathbb{Z}/p\mathbb{Z}$, where $p$ is a prime. A more interesting example of a simple module is the (left) module of complex numbers over the ring $\mathbb{C}\langle x \rangle$ of complex numbers with a noncommuting indeterminate $x$ adjoined, where $x$ corresponds to complex conjugation.