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Prime ideal

In abstract algebra, prime ideals are important generalizations 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:
  • P is not equal to R
  • 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.
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 natural number n is a prime number if and only if the ideal Zn 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 a subset of exactly 2 ideals (which must then be M itself and the entire ring R). Every maximal ideal is in fact prime.
  • 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.

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 commutative ring ≠ 0 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 alternatively, if and only if {0} is a maximal ideal.

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, since it made comprehensible the failure of the fundamental theorem of arithmetic.



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