Maximal Ideal - Generalization

Generalization

For an R module A, a maximal submodule M of A is a submodule M≠A for which for any other submodule N, if M⊆N⊆A then N=M or N=A. Equivalently, M is a maximal submodule if and only if the quotient module A/M is a simple module. Clearly the maximal right ideals of a ring R are exactly the maximal submodules of the module R_R.

Unlike rings with unity however, a 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.

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 by no other proper sub-bimodule of M. So, the maximal ideals of R are exactly the maximal sub-bimodules of the bimodule _RR_R.