Ideal (ring theory)

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any other integer results in another even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring similarly to the way that, in group theory, a normal subgroup can be used to construct a quotient group.

Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may be distinct from the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements of the ring. For instance, the prime ideals of a ring are analogous to prime numbers, and the Chinese remainder theorem can be generalized to ideals. There is a version of unique prime factorization for the ideals of a Dedekind domain (a type of ring important in number theory).

The concept of an order ideal in order theory is derived from the notion of ideal in ring theory. A fractional ideal is a generalization of an ideal, and the usual ideals are sometimes called integral ideals for clarity.


Ideals were first proposed by Richard Dedekind in 1876 in the third edition of his book Vorlesungen über Zahlentheorie (English: Lectures on Number Theory). They were a generalization of the concept of ideal numbers developed by Ernst Kummer.[1][2] Later the concept was expanded by David Hilbert and especially Emmy Noether.


For an arbitrary ring , let be its additive group. A subset is called a two-sided ideal (or simply an ideal) of if it is an additive subgroup of R that "absorbs multiplication by elements of R." Formally we mean that is an ideal if it satisfies the following conditions:

  1. is a subgroup of

Equivalently, an ideal of R is a sub-R-bimodule of R.

A subset of is called a right ideal of [3] if it is an additive subgroup of R and absorbs multiplication on the right, that is:

  1. is a subgroup of

Equivalently, a right ideal of is a right -submodule of .

Similarly a subset of is called a left ideal of if it is an additive subgroup of R absorbing multiplication on the left:

  1. is a subgroup of

Equivalently, a left ideal of is a left -submodule of .

In all cases, the first condition can be replaced by the following well-known criterion that ensures a nonempty subset of a group is a subgroup:

1'. is non-empty and .[4]

The left ideals in R are exactly the right ideals in the opposite ring Ro and vice versa. A two-sided ideal is a left ideal that is also a right ideal, and is often called an ideal except to emphasize that there might exist single-sided ideals. When R is a commutative ring, the definitions of left, right, and two-sided ideal coincide, and the term ideal is used alone.


{0} and R are ideals in every ring R. If R is a division ring or a field, then these are its only ideals. The ideal R is called the unit ideal. I is a proper ideal if it is a proper subset of R, that is, I does not equal R.[5]

Just as normal subgroups of groups are kernels of group homomorphisms, ideals have interpretations as kernels. For a nonempty subset A of R:

If p is in R, then pR is a right ideal and Rp is a left ideal of R. These are called, respectively, the principal right and left ideals generated by p. To remember which is which, note that right ideals are stable under right-multiplication (IR  I) and left ideals are stable under left-multiplication (RI  I).

The connection between cosets and ideals can be seen by switching the operation from "multiplication" to "addition".


Intuitively, the definition can be motivated as follows: Suppose we have a subset of elements Z of a ring R and that we would like to obtain a ring with the same structure as R, except that the elements of Z should be zero (they are in some sense "negligible").

But if and in our new ring, then surely should be zero too, and as well as should be zero for any element (zero or not).

The definition of an ideal is such that the ideal I generated (see below) by Z is exactly the set of elements that are forced to become zero if Z becomes zero, and the quotient ring R/I is the desired ring where Z is zero, and only elements that are forced by Z to be zero are zero. The requirement that R and R/I should have the same structure (except that I becomes zero) is formalized by the condition that the projection from R to R/I is a (surjective) ring homomorphism.


Ideal generated by a set

Let R be a (possibly not unital) ring. Any intersection of any nonempty family of left ideals of R is again a left ideal of R. If X is any subset of R, then the intersection of all left ideals of R containing X is a left ideal I of R containing X, and is clearly the smallest left ideal to do so. This ideal I is said to be the left ideal generated by X. Similar definitions can be created by using right ideals or two-sided ideals in place of left ideals.

If R has unity, then the left, right, or two-sided ideal of R generated by a subset X of R can be expressed internally as we will now describe. The following set is a left ideal:

Each element described would have to be in every left ideal containing X, so this left ideal is in fact the left ideal generated by X. The right ideal and ideal generated by X can also be expressed in the same way:

The former is the right ideal generated by X, and the latter is the ideal generated by X.

By convention, 0 is viewed as the sum of zero such terms, agreeing with the fact that the ideal of R generated by ∅ is {0} by the previous definition.

If a left ideal I of R has a finite subset F such that I is the left ideal generated by F, then the left ideal I is said to be finitely generated. Similar terms are also applied to right ideals and two-sided ideals generated by finite subsets.

In the special case where the set X is just a singleton {a} for some a in R, then the above definitions turn into the following:

These ideals are known as the left/right/two-sided principal ideals generated by a. It is also very common to denote the two-sided ideal generated by a as (a).

If R does not have a unit, then the internal descriptions above must be modified slightly. In addition to the finite sums of products of things in X with things in R, we must allow the addition of n-fold sums of the form x+x+...+x, and n-fold sums of the form (−x)+(−x)+...+(−x) for every x in X and every n in the natural numbers. When R has a unit, this extra requirement becomes superfluous.


Types of ideals

To simplify the description all rings are assumed to be commutative. The non-commutative case is discussed in detail in the respective articles.

Ideals are important because they appear as kernels of ring homomorphisms and allow one to define factor rings. Different types of ideals are studied because they can be used to construct different types of factor rings.

Two other important terms using "ideal" are not always ideals of their ring. See their respective articles for details:

Further properties

Ideal operations

The sum and product of ideals are defined as follows. For and , ideals of a ring R,


i.e. the product of two ideals and is defined to be the ideal generated by all products of the form ab with a in and b in . The product is contained in the intersection of and .

The sum and the intersection of ideals is again an ideal; with these two operations as join and meet, the set of all ideals of a given ring forms a complete modular lattice. Also, the union of two ideals is a subset of the sum of those two ideals, because for any element a inside an ideal, we can write it as a+0, or 0+a, therefore, it is contained in the sum as well. However, the union of two ideals is not necessarily an ideal.

Ideals and congruence relations

There is a bijective correspondence between ideals and congruence relations (equivalence relations that respect the ring structure) on the ring:

Given an ideal I of a ring R, let x ~ y if xyI. Then ~ is a congruence relation on R.

Conversely, given a congruence relation ~ on R, let I = {x : x ~ 0}. Then I is an ideal of R.

See also


  1. Harold M. Edwards (1977). Fermat's last theorem. A genetic introduction to algebraic number theory. p. 76.
  2. Everest G., Ward T. (2005). An introduction to number theory. p. 83.
  3. See Hazewinkel et al. (2004), p. 4.
  4. In fact, since R is assumed to be unital, it suffices that x + y is in I, since the second condition implies that y is in I.
  5. Lang 2005, Section III.2
  6. Because simple commutative rings are fields. See Lam (2001). A First Course in Noncommutative Rings. p. 39.
  • Lang, Serge (2005). Undergraduate Algebra (Third ed.). Springer-Verlag. ISBN 978-0-387-22025-3 
  • Michiel Hazewinkel, Nadiya Gubareni, Nadezhda Mikhaĭlovna Gubareni, Vladimir V. Kirichenko. Algebras, rings and modules. Volume 1. 2004. Springer, 2004. ISBN 1-4020-2690-0
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