Page "Simple module" Paragraph 5

Every simple R-module is isomorphic to a quotient R / m where m is a maximal right ideal of R. By the above paragraph, any quotient R / m is a simple module.

Conversely, suppose that M is a simple R-module.

Then, for any non-zero element x of M, the cyclic submodule xR must equal M. Fix such an x.

The statement that xR = M is equivalent to the surjectivity of the homomorphism that sends r to xr.

The kernel of this homomorphism is a right ideal I of R, and a standard theorem states that M is isomorphic to R / I.

By the above paragraph, we find that I is a maximal right ideal.

Therefore M is isomorphic to a quotient of R by a maximal right ideal.