Inmathematics, more specifically inring theory, amaximal ideal is anideal that ismaximal (with respect toset inclusion) amongst allproper ideals.^[1][2] In other words,I is a maximal ideal of aringR if there are no other ideals contained betweenI andR.

Maximal ideals are important because thequotients of rings by maximal ideals aresimple rings, and in the special case ofunitalcommutative rings they are alsofields.

In noncommutative ring theory, amaximal right ideal is defined analogously as being a maximal element in theposet of proper right ideals, and similarly, amaximal left ideal is defined to be a maximal element of the poset of proper left ideals. Since a one-sided maximal idealA is not necessarily two-sided, the quotientR/A is not necessarily a ring, but it is asimple module overR. IfR has a unique maximal right ideal, thenR is known as alocal 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 theJacobson radical J(R).

It is possible for a ring to have a unique maximal two-sided ideal and yet lack unique maximal one-sided ideals: for example, in the ring of 2 by 2square matrices over a field, thezero ideal is a maximal two-sided ideal, but there are many maximal right ideals.

There are other equivalent ways of expressing the definition of maximal one-sided and maximal two-sided ideals. Given a ringR and a proper idealI ofR (that isI ≠R),I is a maximal ideal ofR if any of the following equivalent conditions hold:

• There exists no other proper idealJ ofR so thatI ⊊J.
• For any idealJ withI ⊆J, eitherJ =I orJ =R.
• The quotient ringR/I is a simple ring.

There is an analogous list for one-sided ideals, for which only the right-hand versions will be given. For a right idealA of a ringR, the following conditions are equivalent toA being a maximal right ideal ofR:

• There exists no other proper right idealB ofR so thatA ⊊B.
• For any right idealB withA ⊆B, eitherB =A orB =R.
• The quotient moduleR/A is a simple rightR-module.

Maximal right/left/two-sided ideals are thedual notion to that ofminimal ideals.

• IfF is a field, then the only maximal ideal is {0}.
• In the ringZ of integers, the maximal ideals are theprincipal ideals generated by a prime number.
• More generally, all nonzeroprime ideals are maximal in aprincipal ideal domain.
• The ideal${\displaystyle (2,x)}$  is a maximal ideal in ring${\displaystyle \mathbb{Z}[x]}$ . Generally, the maximal ideals of${\displaystyle \mathbb{Z}[x]}$  are of the form${\displaystyle (p,f(x))}$  where${\displaystyle p}$  is a prime number and${\displaystyle f(x)}$  is a polynomial in${\displaystyle \mathbb{Z}[x]}$  which is irreducible modulo${\displaystyle p}$ .
• Every prime ideal is a maximal ideal in a Boolean ring, i.e., a ring consisting of only idempotent elements. In fact, every prime ideal is maximal in a commutative ring${\displaystyle R}$  whenever there exists an integer${\displaystyle n>1}$  such that${\displaystyle x^n=x}$  for any${\displaystyle x\in R}$ .
• The maximal ideals of thepolynomial ring${\displaystyle \mathbb{C}[x]}$  are principal ideals generated by${\displaystyle x-c}$  for some${\displaystyle c\in \mathbb{C}}$ .
• More generally, the maximal ideals of the polynomial ringK[x₁, ...,x_n] over analgebraically closed fieldK are the ideals of the form(x₁ − a₁, ...,x_n − a_n). This result is known as the weakNullstellensatz.

• An important ideal of the ring called theJacobson radical can be defined using maximal right (or maximal left) ideals.
• IfR is a unital commutative ring with an idealm, thenk =R/m is a field if and only ifm is a maximal ideal. In that case,R/m is known as theresidue field. This fact can fail in non-unital rings. For example,${\displaystyle 4\mathbb{Z}}$  is a maximal ideal in${\displaystyle 2\mathbb{Z}}$ , but${\displaystyle 2\mathbb{Z}/4\mathbb{Z}}$  is not a field.
• IfL is a maximal left ideal, thenR/L is a simple leftR-module. Conversely in rings with unity, any simple leftR-module arises this way. Incidentally this shows that a collection of representatives of simple leftR-modules is actually a set since it can be put into correspondence with part of the set of maximal left ideals ofR.
• Krull's theorem (1929): Every nonzero unital ring 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 nonzerofinitely generated module has a maximal submodule. SupposeI is an ideal which is notR (respectively,A is a right ideal which is notR). ThenR/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) ofR containingI (respectively,A).
• Krull's theorem can fail for rings without unity. Aradical ring, i.e. a ring in which theJacobson radical is the entire ring, has no simple modules and hence has no maximal right or left ideals. Seeregular ideals for possible ways to circumvent this problem.
• In a commutative ring with unity, every maximal ideal is aprime ideal. The converse is not always true: for example, in any nonfieldintegral domain the zero ideal is a prime ideal which is not maximal. Commutative rings in which prime ideals are maximal are known aszero-dimensional rings, where the dimension used is theKrull dimension.
• A maximal ideal of a noncommutative ring might not be prime in the commutative sense. For example, let${\displaystyle M_{n\times n}(\mathbb{Z})}$  be the ring of all${\displaystyle n\times n}$  matrices over${\displaystyle \mathbb{Z}}$ . This ring has a maximal ideal${\displaystyle M_{n\times n}(p\mathbb{Z})}$  for any prime${\displaystyle p}$ , but this is not a prime ideal since (in the case${\displaystyle n=2}$ )${\displaystyle A=\text{diag}(1,p)}$  and${\displaystyle B=\text{diag}(p,1)}$  are not in${\displaystyle M_{n\times n}(p\mathbb{Z})}$ , but${\displaystyle AB=p{I}_2\in M_{n\times n}(p\mathbb{Z})}$ . However, maximal ideals of noncommutative ringsare prime in thegeneralized sense below.

For anR-moduleA, amaximal submoduleM ofA is a submoduleM ≠A satisfying the property that for any other submoduleN,M ⊆N ⊆A impliesN =M orN =A. Equivalently,M is a maximal submodule if and only if the quotient moduleA/M is asimple module. The maximal right ideals of a ringR are exactly the maximal submodules of the moduleR_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 alsoprojective modules have maximal submodules.

As with rings, one can define theradical of a module using maximal submodules. Furthermore, maximal ideals can be generalized by defining amaximal sub-bimoduleM of abimoduleB to be a proper sub-bimodule ofM which is contained in no other proper sub-bimodule ofM. The maximal ideals ofR are then exactly the maximal sub-bimodules of the bimodule_RR_R.

• Prime ideal

• Anderson, Frank W.; Fuller, Kent R. (1992),Rings and categories of modules, Graduate Texts in Mathematics, vol. 13 (2 ed.), New York: Springer-Verlag, pp. x+376,doi:10.1007/978-1-4612-4418-9,ISBN 0-387-97845-3,MR 1245487
• Lam, T. Y. (2001),A first course in noncommutative rings, Graduate Texts in Mathematics, vol. 131 (2 ed.), New York: Springer-Verlag, pp. xx+385,doi:10.1007/978-1-4419-8616-0,ISBN 0-387-95183-0,MR 1838439