In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. Prime ideals have a simpler description for commutative rings, so we consider this case separately below. This article only covers ideals of ring theory. Prime ideals in order theory are treated in the article on ideals in order theory. Main article: History of mathematics The evolution of mathematics can be seen to be an ever increasing series of abstractions. ... A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ... In mathematical order theory, an ideal is a special subset of a partially ordered set. ...

Prime ideals for commutative rings

If R is a commutative ring, then an ideal P of R is prime if it has the following two properties: In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ...

• whenever a, b are two elements of R such that their product ab lies in P, then a is in P or b is in P.
• P is not equal to the whole ring R

This generalizes the following property of prime numbers: if p is a prime number and if p divides a product ab of two integers, then p divides a or p divides b. We can therefore say The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...

A positive integer n is a prime number if and only if the ideal nZ is a prime ideal in Z.

Examples

• If R denotes the ring C[X, Y] of polynomials in two variables with complex coefficients, then the ideal generated by the polynomial Y² − X³ − X − 1 is a prime ideal (see elliptic curve).
• In the ring Z[X] of all polynomials with integer coefficients, the ideal generated by 2 and X is a prime ideal. It consists of all those polynomials whose constant coefficient is even.
• In any ring R, a maximal ideal is an ideal M that is maximal in the set of all proper ideals of R, i.e. M is a contained in exactly 2 ideals of R, namely M itself and the entire ring R. Every maximal ideal is in fact prime; the converse is not true, in general.
• If M is a smooth manifold, R is the ring of smooth functions on M, and x is a point in M, then the set of all smooth functions f with f(x) = 0 forms a prime ideal (even a maximal ideal) in R.

In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ... In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections. ... In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set is an element of S that is not smaller than any other element in S. The term minimal element is defined dually. ... A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X... This is the current mathematics collaboration of the week! Please help improve it to featured article standard at manifold/rewrite. ...

Properties

• An ideal I in the commutative ring R is prime if and only if the factor ring R/I is an integral domain.
• Every maximal ideal (see above) is prime; an ideal I in the commutative ring R is a maximal ideal if and only if the factor ring R/I is a field.
• Every nonzero commutative ring contains at least one prime ideal. In fact, it contains at least one maximal ideal, which can be proven using Zorn's lemma.
• A commutative ring is an integral domain if and only if {0} is a prime ideal.
• A commutative ring is a field if and only if {0} is its only prime ideal, or equivalently, if and only if {0} is a maximal ideal.

In abstract algebra, an integral domain is a commutative ring with 0 ≠ 1 in which the product of any two non-zero elements is always non-zero. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... Zorns lemma, also known as the Kuratowski-Zorn lemma, is a theorem of set theory that states: Every partially ordered set in which every chain (i. ... In abstract algebra, an integral domain is a commutative ring with 0 ≠ 1 in which the product of any two non-zero elements is always non-zero. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

Uses

One use of prime ideals occurs in algebraic geometry, where varieties are defined as the zero sets of ideals in polynomial rings. It turns out that the irreducible varieties correspond to prime ideals. In the modern abstract approach, one starts with an arbitrary commutative ring and turns the set of its prime ideals, also called its spectrum, into a topological space and can thus define generalizations of varieties called schemes, which find applications not only in geometry, but also in number theory. Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... Geometry (from the Greek words Geo = earth and metro = measure) is the branch of mathematics first popularized in ancient Greek culture by Thales (circa 624-547 BC) dealing with spatial relationships. ... Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. ...

The introduction of prime ideals in algebraic number theory was a major step forward: it was realized that the important property of unique factorisation expressed in the fundamental theorem of arithmetic does not hold in every ring of algebraic integers, but a substitute was found when Dedekind replaced elements by ideals and prime elements by prime ideals; see Dedekind domain. In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days... In mathematics, and in particular number theory, the fundamental theorem of arithmetic or unique factorization theorem is the statement that every positive integer greater than 1 can be written as a product of prime numbers in only one way. ... In mathematics, an algebraic integer is a complex number α that is a root of an equation P(x) = 0 where P(x) is a monic polynomial (that is, the coefficient of the largest power of x in P(x) is one) with integer coefficients. ... Julius Wilhelm Richard Dedekind (October 6, 1831 - February 12, 1916) was a German mathematician and Ernst Eduard Kummers closest follower in arithmetic. ... In abstract algebra, a Dedekind domain is a Noetherian integral domain which is integrally closed in its fraction field and which has Krull dimension 1. ...

Prime ideals for noncommutative rings

If R is a noncommutative ring, then an ideal P of R is prime if it has the following two properties: In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ...

• whenever a, b are two elements of R such that for all elements r of R, their product arb lies in P, then a is in P or b is in P.
• P is not equal to the whole ring R.

For commutative rings this definition is equivalent to the one given in the previous section. For noncommutative rings, the two definitions are different. An ideal such that ab in P implies that a or b is in P is called a completely prime ideal. Completely prime ideals are prime ideals, but the converse is not true. For example, the zero ideal in the ring of n × n matrices is a prime ideal, but it is not completely prime.

Examples

• Any maximal ideal is prime.

• Any primitive ideal is prime.

• The zero ideal of any prime ring is prime.

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... A left primitive ideal is the annihilator of a simple left module. ... In abstract algebra, a ring R is a prime ring if for any two elements a and b of R, if arb = 0 for all r in R, then either a = 0 or b = 0. ...

Properties

• An ideal P is prime if and only if for two ideals A and B, AB = P implies that either A or B is contained in P. This is sometimes taken as the definition of a prime ideal. This is close to the historical point of view of ideals as ideal numbers, as "A is contained in P" is another way of saying "P divides A".