In noncommutative ring theory, a maximal right ideal is defined analogously as being a maximal element in the poset of proper right ideals, and similarly, a maximal left ideal is defined to be a maximal element of the poset of proper left ideals.
Since a one-sided maximal ideal A is not necessarily two-sided, the quotient R/A is not necessarily a ring, but it is a simple module over R. If R has a unique maximal right ideal, then R is known as a local ring, and the maximal right ideal is also the unique maximal left and unique maximal two-sided ideal of the ring, and is in fact the Jacobson radical J(R).
Unlike rings with unity, a nonzero module does not necessarily have maximal submodules.
However, as noted above, finitely generated nonzero modules have maximal submodules, and also projective modules have maximal submodules.
As with rings, one can define the radical of a module using maximal submodules.