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Prime ideal

In abstract algebra, the concept of prime ideals is an important generalization of the concept of prime numbers. If R is a commutative ring, then an ideal P of R is called prime if it has the following two properties: 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
A positive integer n is a prime number if and only if the ideal Zn is a prime ideal in Z.

Table of contents
1 Examples
2 Properties
3 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.

The introduction of prime ideals in algebraic number theory was a major step forward: it was realized that the ordinary fundamental theorem of arithmetic does not work in rings of algebraic integers, but a substitute was found when Dedekind replaced elements by ideals and prime elements by prime ideals; see Dedekind domain.