to the quotient to conclude that there is a maximal ideal (respectively, maximal right ideal) of R containing I (respectively, A). Krull's theorem can fail for rings without unity. A radical ring, i.e. a ring in which the Jacobson radical is the entire ring, has no simple modules and hence has no maximal right or left ideals. See regular ideals for possible ways to circumvent this problem. In a commutative ring with unity, every maximal ideal is a prime ideal. The converse is not always true: for example, in any nonfield integral domain the zero ideal is a prime ideal which is not maximal. Commutative rings in which prime ideals are maximal are known as zero-dimensional rings, where the dimension used is the Krull dimension. A maximal ideal of a noncommutative ring might not be prime in the commutative sense. For example, let be the ring of all matrices over . This ring has a maximal ideal for any prime , but this is not a prime ideal since (in the case ) and are not in , but . However, maximal ideals of noncommutative rings are prime in the generalized sense below. Generalization For an R-module A, a maximal submodule M of A is a submodule satisfying the property that for any other submodule N, implies or . 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 RR. 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. 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 RRR. See also Prime ideal References Ideals (ring theory) Ring theory Prime ideals