Ideal (ring theory)

In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements. 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 integer (even or odd) results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, 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 not correspond directly to 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 related, but distinct, concept of an ideal in order theory is derived from the notion of an ideal in ring theory. A fractional ideal is a generalization of an ideal, and the usual ideals are sometimes called integral ideals for clarity.

Ernst Kummer invented the concept of ideal numbers to serve as the "missing" factors in number rings in which unique factorization fails; here the word "ideal" is in the sense of existing in imagination only, in analogy with "ideal" objects in geometry such as points at infinity. In 1876, Richard Dedekind replaced Kummer's undefined concept by concrete sets of numbers, sets that he called ideals, in the third edition of Dirichlet's book Vorlesungen über Zahlentheorie, to which Dedekind had added many supplements. Later the notion was extended beyond number rings to the setting of polynomial rings and other commutative rings by David Hilbert and especially Emmy Noether.

Given a ring \(R\), a left ideal is a subset \(I\) of \(R\) that is a subgroup of the additive group of \(R\) that is closed under left multiplication by elements of ⁠\(R\)⁠; that is, \(0 \in I\) and for every \(r \in R\) and every ⁠\(x, y \in I\)⁠, one has

• ⁠\(x+y\in I\)⁠
• ⁠\(-x\in I\)⁠
• ⁠\(rx\in I\)⁠.

In other words, a left ideal is a left submodule of \(R\), considered as a left module over itself.

A right ideal is defined similarly, with the condition \(rx\in I\) replaced by ⁠\(xr\in I\)⁠. A two-sided ideal is a left ideal that is also a right ideal.

If the ring is commutative, the definitions of left, right, and two-sided ideal coincide, and one talks simply of an ideal. In the non-commutative case, "ideal" is often used instead of "two-sided ideal".

Since an ideal \(I\) is an abelian subgroup, the relation between ⁠\(x\)⁠ and ⁠\(y\)⁠ defined by

\(x-y\in I\)

is an equivalence relation on \(R\), and the set of equivalence classes is an abelian group denoted ⁠\(R/I\)⁠ and called the quotient of \(R\) by \(I\). If \(I\) is a left or a right ideal, the quotient ⁠\(R/I\)⁠ is a left or right ⁠\(R\)⁠-module, respectively.

If the ideal \(I\) is two-sided, the quotient \(R/I\) is a ring, and the function

\(R\to R/I\)

Condensed: the full section is in Wikipedia.

(For the sake of brevity, some results are stated only for left ideals but are usually also true for right ideals with appropriate notation changes.)

Condensed: the full section is in Wikipedia.

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.

• Maximal ideal: A proper ideal I is called a maximal ideal if there exists no other proper ideal J with I a proper subset of J. The factor ring of a maximal ideal is a simple ring in general and is a field for commutative rings.
• Minimal ideal: A nonzero ideal is called minimal if it contains no other nonzero ideal.
• Zero ideal: the ideal \(\{0\}\).
• Unit ideal: the whole ring (being the ideal generated by \(1\)).
• Prime ideal: A proper ideal \(I\) is called a prime ideal if for any \(a\) and \(b\) in ⁠\(R\)⁠, if \(ab\) is in ⁠\(I\)⁠, then at least one of \(a\) and \(b\) is in ⁠\(I\)⁠. The factor ring of a prime ideal is a prime ring in general and is an integral domain for commutative rings.
• Radical ideal or semiprime ideal: A proper ideal I is called radical or semiprime if for any a in \(R\), if a is in I for some n, then a is in I. The factor ring of a radical ideal is a semiprime ring for general rings, and is a reduced ring for commutative rings.
• Primary ideal: An ideal I is called a primary ideal if for all a and b in R, if ab is in I, then at least one of a and b is in I for some natural number n. Every prime ideal is primary, but not conversely. A semiprime primary ideal is prime.
• Principal ideal: An ideal generated by one element.
• Finitely generated ideal: This type of ideal is finitely generated as a module.
• Primitive ideal: A left primitive ideal is the annihilator of a simple left module.
• Irreducible ideal: An ideal is said to be irreducible if it cannot be written as an intersection of ideals that properly contain it.
• Comaximal ideals: Two ideals I, J are said to be comaximal if \(x + y = 1\) for some \(x \in I\) and ⁠\(y \in J\)⁠.
• Regular ideal: This term has multiple uses. See the article for a list.
• Nil ideal: An ideal is a nil ideal if each of its elements is nilpotent.
• Nilpotent ideal: Some power of it is zero.
• Parameter ideal: an ideal generated by a system of parameters.
• Perfect ideal: A proper ideal I in a Noetherian ring \(R\) is called a perfect ideal if its grade equals the projective dimension of the associated quotient ring, ⁠\(\textrm{grade}(I)=\textrm{proj}\dim(R/I)\)⁠. A perfect ideal is unmixed.
• Unmixed ideal: A proper ideal I in a Noetherian ring \(R\) is called an unmixed ideal (in height) if the height of I is equal to the height of every associated prime P of \(R/I\). (This is stronger than saying that \(R/I\) is equidimensional. See also equidimensional ring.

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

Condensed: the full section is in Wikipedia.

The sum and product of ideals are defined as follows. For \(\mathfrak{a}\) and ⁠\(\mathfrak{b}\)⁠, left (resp. right) ideals of a ring R, their sum is

\(\mathfrak{a}+\mathfrak{b}:=\{a+b \mid a \in \mathfrak{a} \mbox{ and } b \in \mathfrak{b}\},\)

which is a left (resp. right) ideal, and, if \(\mathfrak{a}, \mathfrak{b}\) are two-sided,

\(\mathfrak{a} \mathfrak{b}:=\{a_1b_1+ \dots + a_nb_n \mid a_i \in \mathfrak{a} \mbox{ and } b_i \in \mathfrak{b}, i=1, 2, \dots, n; \mbox{ for } n=1, 2, \dots\},\)

i.e. the product is the ideal generated by all products of the form ab with a in \(\mathfrak{a}\) and b in ⁠\(\mathfrak{b}\)⁠.

Note \(\mathfrak{a} + \mathfrak{b}\) is the smallest left (resp. right) ideal containing both \(\mathfrak{a}\) and \(\mathfrak{b}\) (or the union ⁠\(\mathfrak{a} \cup \mathfrak{b}\)⁠), while the product \(\mathfrak{a}\mathfrak{b}\) is contained in the intersection of \(\mathfrak{a}\) and ⁠\(\mathfrak{b}\)⁠.

The distributive law holds for two-sided ideals ⁠\(\mathfrak{a}, \mathfrak{b}, \mathfrak{c}\)⁠,

• ⁠\(\mathfrak{a}(\mathfrak{b} + \mathfrak{c}) = \mathfrak{a} \mathfrak{b} + \mathfrak{a} \mathfrak{c}\)⁠,
• ⁠\((\mathfrak{a} + \mathfrak{b}) \mathfrak{c} = \mathfrak{a}\mathfrak{c} + \mathfrak{b}\mathfrak{c}\)⁠.

If a product is replaced by an intersection, a partial distributive law holds:

\(\mathfrak{a} \cap (\mathfrak{b} + \mathfrak{c}) \supset \mathfrak{a} \cap \mathfrak{b} + \mathfrak{a} \cap \mathfrak{c}\)

where the equality holds if \(\mathfrak{a}\) contains \(\mathfrak{b}\) or \(\mathfrak{c}\).

• \(\mathfrak{a} + \mathfrak{b} = (1)\)
• \(\mathfrak{a}\) is generated by elements that form a regular sequence modulo ⁠\(\mathfrak{b}\)⁠.

Condensed: the full section is in Wikipedia.

In \(\mathbb{Z}\) we have

\((n)\cap(m) = \operatorname{lcm}(n,m)\mathbb{Z}\)

since \((n)\cap(m)\) is the set of integers that are divisible by both \(n\) and ⁠\(m\)⁠.

Let \(R = \mathbb{C}[x,y,z,w]\) and let ⁠\(\mathfrak{a} = (z, w), \mathfrak{b} = (x+z,y+w),\mathfrak{c} = (x+z, w)\)⁠. Then,

• \(\mathfrak{a} + \mathfrak{b} = (z,w, x+z, y+w) = (x, y, z, w)\) and \(\mathfrak{a} + \mathfrak{c} = (z, w, x)\)
• \(\mathfrak{a}\mathfrak{b} = (z(x + z), z(y + w), w(x + z), w(y + w))= (z^2 + xz, zy + wz, wx + wz, wy + w^2)\)
• \(\mathfrak{a}\mathfrak{c} = (xz + z^2, zw, xw + zw, w^2)\)
• \(\mathfrak{a} \cap \mathfrak{b} = \mathfrak{a}\mathfrak{b}\) while \(\mathfrak{a} \cap \mathfrak{c} = (w, xz + z^2) \neq \mathfrak{a}\mathfrak{c}\)

In the first computation, we see the general pattern for taking the sum of two finitely generated ideals, it is the ideal generated by the union of their generators. In the last three we observe that products and intersections agree whenever the two ideals intersect in the zero ideal. These computations can be checked using Macaulay2.

Ideals appear naturally in the study of modules, especially in the form of a radical.

For simplicity, we work with commutative rings but, with some changes, the results are also true for non-commutative rings.

Let R be a commutative ring. By definition, a primitive ideal of R is the annihilator of a (nonzero) simple R-module. The Jacobson radical \(J = \operatorname{Jac}(R)\) of R is the intersection of all primitive ideals. Equivalently,

\(J = \bigcap_{\mathfrak{m} \text{ maximal ideals}} \mathfrak{m}.\)

Indeed, if \(M\) is a simple module and x is a nonzero element in M, then \(Rx = M\) and \(R/\operatorname{Ann}(M) = R/\operatorname{Ann}(x) \simeq M\), meaning \(\operatorname{Ann}(M)\) is a maximal ideal. Conversely, if \(\mathfrak{m}\) is a maximal ideal, then \(\mathfrak{m}\) is the annihilator of the simple R-module ⁠\(R/\mathfrak{m}\)⁠. There is also another characterization (the proof is not hard):

\(J = \{ x \in R \mid 1 - yx \, \text{ is a unit element for every } y \in R\}.\)

For a not-necessarily-commutative ring, it is a general fact that \(1 - yx\) is a unit element if and only if \(1 - xy\) is (see the link) and so this last characterization shows that the radical can be defined both in terms of left and right primitive ideals.

The following simple but important fact (Nakayama's lemma) is built-in to the definition of a Jacobson radical: if M is a module such that ⁠\(JM = M\)⁠, then M does not admit a maximal submodule, since if there is a maximal submodule ⁠\(L \subsetneq M\)⁠, \(J \cdot (M/L) = 0\) and so ⁠\(M = JM \subset L \subsetneq M\)⁠, a contradiction. Since a nonzero finitely generated module admits a maximal submodule, in particular, one has:

If \(JM = M\) and M is finitely generated, then ⁠\(M = 0\)⁠.

A maximal ideal is a prime ideal and so one has

\(\operatorname{nil}(R) = \bigcap_{\mathfrak{p} \text { prime ideals }} \mathfrak{p} \subset \operatorname{Jac}(R)\)

where the intersection on the left is called the nilradical of R. As it turns out, \(\operatorname{nil}(R)\) is also the set of nilpotent elements of R.

Condensed: the full section is in Wikipedia.

Let \(A\) and \(B\) be two commutative rings, and let \(f: A\to B\) be a ring homomorphism. If \(\mathfrak{a}\) is an ideal in \(A\), then \(f(\mathfrak{a})\) need not be an ideal in \(B\) (e.g. take \(f\) to be the inclusion of the ring of integers \(\mathbb{Z}\) into the field of rationals \(\mathbb{Q}\)). The extension \(\mathfrak{a}^e\) of \(\mathfrak{a}\) in \(B\) is defined to be the ideal in \(B\) generated by ⁠\(f(\mathfrak{a})\)⁠. Explicitly,

\(\mathfrak{a}^e = f(\mathfrak{a})B = \Big\{ \sum y_if(x_i) : x_i \in \mathfrak{a}, y_i \in B \Big\}.\)

By abuse of notation, \(\mathfrak{a}B\) is another common notation for this ideal extension.

If \(\mathfrak{b}\) is an ideal of \(B\), then \(f^{-1}(\mathfrak{b})\) is always an ideal of \(A\), called the contraction \(\mathfrak{b}^c\) of \(\mathfrak{b}\) to \(A\).

Assuming \(f: A\to B\) is a ring homomorphism, \(\mathfrak{a}\) is an ideal in \(A\), \(\mathfrak{b}\) is an ideal in \(B\), then:

• \(\mathfrak{b}\) is prime in \(B\) \(\Rightarrow\) \(\mathfrak{b}^c\) is prime in \(A\),
• \(\mathfrak{a}^{ec} \supseteq \mathfrak{a},\)
• \(\mathfrak{b}^{ce} \subseteq \mathfrak{b}.\)

It is false, in general, that \(\mathfrak{a}\) being prime (or maximal) in \(A\) implies that \(\mathfrak{a}^e\) is prime (or maximal) in \(B\). Many classic examples of this stem from algebraic number theory. For example, consider the embedding \(\mathbb{Z} \to \mathbb{Z}\left\lbrack i \right\rbrack.\) In \(B = \mathbb{Z}\left\lbrack i \right\rbrack\), the element 2 factors as \(2 = (1 + i)(1 - i)\) where (one can show) neither of \(1 + i, 1 - i\) are units in \(B\). So \((2)^e\) is not prime in \(B\) (and therefore not maximal, as well). Indeed, \((1 \pm i)^2 = \pm 2i\) shows that ⁠\((1 + i) = ((1 - i) - (1 - i)^2)\)⁠, ⁠\((1 - i) = ((1 + i) - (1 + i)^2)\)⁠, and therefore ⁠\((2)^e = (1 + i)^2\)⁠.

On the other hand, if \(f\) is surjective and \(\mathfrak{a} \supseteq \ker f\) then:

• \(\mathfrak{a}^{ec}=\mathfrak{a}\) and \(\mathfrak{b}^{ce}=\mathfrak{b},\)
• \(\mathfrak{a}\) is a prime ideal in \(A\) \(\Leftrightarrow\) \(\mathfrak{a}^e\) is a prime ideal in \(B\),
• \(\mathfrak{a}\) is a maximal ideal in \(A\) \(\Leftrightarrow\) \(\mathfrak{a}^e\) is a maximal ideal in \(B\).

Remark: Let \(K\) be a field extension of \(L\), and let \(B\) and \(A\) be the rings of integers of \(K\) and \(L\), respectively. Then \(B\) is an integral extension of \(A\), and we let \(f\) be the inclusion map from \(A\) to \(B\). The behaviour of a prime ideal \(\mathfrak{a} = \mathfrak{p}\) of \(A\) under extension is one of the central problems of algebraic number theory.

Condensed: the full section is in Wikipedia.

Ideals can be generalized to any monoid object ⁠\((R,\otimes)\)⁠, where \(R\) is the object where the monoid structure has been forgotten. A left ideal of \(R\) is a subobject \(I\) that "absorbs multiplication from the left by elements of ⁠\(R\)⁠"; that is, \(I\) is a left ideal if it satisfies the following two conditions:

1. \(I\) is a subobject of \(R\)
2. For every \(r \in (R,\otimes)\) and every ⁠\(x \in (I, \otimes)\)⁠, the product \(r \otimes x\) is in ⁠\((I, \otimes)\)⁠.

A right ideal is defined with the condition "⁠\(r \otimes x \in (I, \otimes)\)⁠" replaced by "'⁠\(x \otimes r \in (I, \otimes)\)⁠". A two-sided ideal is a left ideal that is also a right ideal, and is sometimes simply called an ideal. When \(R\) is a commutative monoid object respectively, the definitions of left, right, and two-sided ideal coincide, and the term ideal is used alone.