Maximal Ideal - Properties

Properties

• An important ideal of the ring called the Jacobson radical can be defined using maximal right (or maximal left) ideals.

• If R is a unital commutative ring with an ideal m, then k = R/m is a field if and only if m is a maximal ideal. In that case, R/m is known as the residue field. This fact can fail in non-unital rings. For example, is a maximal ideal in, but is not a field.

• If L is a maximal left ideal, then R/L is a simple left R module. Conversely in rings with unity, any simple left R module arises this way. Incidentally this shows that a collection of representatives of simple left R modules is actually a set since it can be put into correspondence with part of the set of maximal left ideals of R.

• Krull's theorem (1929): Every ring with a multiplicative identity has a maximal ideal. The result is also true if "ideal" is replaced with "right ideal" or "left ideal". More generally, it is true that every nonzero finitely generated module has a maximal submodule. Suppose I is an ideal which is not R (respectively, A is a right ideal which is not R). Then R/I is a ring with unity, (respectively, R/A is a finitely generated module), and so the above theorems can be applied 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.