Editing Maximal ideal (section)

==Generalization==
For an ''R''-module ''A'', a '''maximal submodule''' ''M'' of ''A'' is a submodule {{nowrap|''M'' ≠ ''A''}} satisfying the property that for any other submodule ''N'', {{nowrap|''M'' ⊆ ''N'' ⊆ ''A''}} implies {{nowrap|1=''N'' = ''M''}} or {{nowrap|1=''N'' = ''A''}}. Equivalently, ''M'' is a maximal submodule if and only if the quotient module ''A''/''M'' is a [[simple module]]. The maximal right ideals of a ring ''R'' are exactly the maximal submodules of the module ''R''<sub>''R''</sub>.

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 module]]s have maximal submodules.

As with rings, one can define the [[radical of a module]] using maximal submodules. Furthermore, maximal ideals can be generalized by defining a '''maximal sub-bimodule''' ''M'' of a [[bimodule]] ''B'' to be a proper sub-bimodule of ''M'' which is contained in no other proper sub-bimodule of ''M''. The maximal ideals of ''R'' are then exactly the maximal sub-bimodules of the bimodule <sub>''R''</sub>''R''<sub>''R''</sub>.