Principal ideal domain

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Principal ideal domain

In abstract algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as group , ring , field , module , vector spaces, and algebra over a field....
, a principal ideal domain, or PID is an integral domain

Integral domain

In abstract algebra, an integral domain is a commutative ring without zero divisors and with a multiplicative identity 1 not equal to 0, the additive identity....
in which every ideal

Ideal (ring theory)

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring . The ideal concept generalizes in an appropriate way some important properties of integers like "even number" or "multiple of 3"....
is principal

Principal ideal

In ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R....
, i.e., can be generated by a single element.

Principal ideal domains are thus mathematical objects which behave somewhat like the integers, with respect to divisibility

Integral domain

In abstract algebra, an integral domain is a commutative ring without zero divisors and with a multiplicative identity 1 not equal to 0, the additive identity....
: any element of a PID has a unique decomposition into prime elements

Integral domain

In abstract algebra, an integral domain is a commutative ring without zero divisors and with a multiplicative identity 1 not equal to 0, the additive identity....
(so an analogue of the fundamental theorem of arithmetic

Fundamental theorem of arithmetic

In number theory and algebraic number theory, the Fundamental Theorem of Arithmetic states that any integer greater than 1 can be written as a unique product of prime numbers....
holds); any two elements of a PID have a greatest common divisor

Greatest common divisor

In mathematics, the greatest common divisor , sometimes known as the greatest common factor or highest common factor , of two non-zero integers, is the largest positive integer that divisor both numbers without remainder....
.

A principal ideal domain is a specific type of integral domain, and can be characterized by the following (not necessarily exhaustive) chain of class inclusions

Subclass (set theory)

In set theory and its applications throughout mathematics, a subclass is a class contained in some other class in the same way that a subset is a Set contained in some other set....
:

ples include:

Examples of integral domains that are not PIDs: It is not principal because the ideal generated by 2 and X is an example of an ideal that cannot be generated by a single polynomial.

key result here is the structure theorem for finitely generated modules over a principal ideal domain
Structure theorem for finitely generated modules over a principal ideal domain

In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules can be uniquely decomposed in much the same way that integers have a prime f...
.

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Encyclopedia

In abstract algebra
Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as group , ring , field , module , vector spaces, and algebra over a field....
, a principal ideal domain, or PID is an integral domain
Integral domain

In abstract algebra, an integral domain is a commutative ring without zero divisors and with a multiplicative identity 1 not equal to 0, the additive identity....
in which every ideal
Ideal (ring theory)

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring . The ideal concept generalizes in an appropriate way some important properties of integers like "even number" or "multiple of 3"....
is principal
Principal ideal

In ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R....
, i.e., can be generated by a single element.

Principal ideal domains are thus mathematical objects which behave somewhat like the integers, with respect to divisibility
Integral domain

In abstract algebra, an integral domain is a commutative ring without zero divisors and with a multiplicative identity 1 not equal to 0, the additive identity....
: any element of a PID has a unique decomposition into prime elements
Integral domain

In abstract algebra, an integral domain is a commutative ring without zero divisors and with a multiplicative identity 1 not equal to 0, the additive identity....
(so an analogue of the fundamental theorem of arithmetic
Fundamental theorem of arithmetic

In number theory and algebraic number theory, the Fundamental Theorem of Arithmetic states that any integer greater than 1 can be written as a unique product of prime numbers....
holds); any two elements of a PID have a greatest common divisor
Greatest common divisor

In mathematics, the greatest common divisor , sometimes known as the greatest common factor or highest common factor , of two non-zero integers, is the largest positive integer that divisor both numbers without remainder....
.

A principal ideal domain is a specific type of integral domain, and can be characterized by the following (not necessarily exhaustive) chain of class inclusions
Subclass (set theory)

In set theory and its applications throughout mathematics, a subclass is a class contained in some other class in the same way that a subset is a Set contained in some other set....
:

Commutative ring
Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a Ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
s ⊃ integral domain
Integral domain

In abstract algebra, an integral domain is a commutative ring without zero divisors and with a multiplicative identity 1 not equal to 0, the additive identity....
s ⊃ integrally closed domains ⊃ unique factorization domain
Unique factorization domain

In mathematics, a unique factorization domain is, roughly speaking, a commutative ring in which every element, with special exceptions, can be uniquely written as a product of prime elements, analogous to the fundamental theorem of arithmetic for the integers....
s ⊃ principal ideal domain
Principal ideal domain

In abstract algebra, a principal ideal domain, or PID is an integral domain in which every ideal is principal ideal, i.e., can be generated by a single element....
s ⊃ Euclidean domain
Euclidean domain

In abstract algebra, a Euclidean domain is a type of Ring in which the Euclidean algorithm applies. Euclidean domains possess many important properties similar to the integers: for example, the fundamental theorem of arithmetic holds in any Euclidean domain....
s ⊃ field
Field (mathematics)

In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication and division , satisfying certain axioms....
s

Examples
Examples include:
K: any field
Field (mathematics)

In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication and division , satisfying certain axioms....
,
Z: the ring
Ring (mathematics)

In mathematics, a ring is a type of algebraic structure. There is some variation among mathematicians as to exactly what properties a ring is required to have, as described in detail below....
of integers
Integer

The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
,
K[x]: rings of polynomials
Polynomial

In mathematics, a polynomial is an expression constructed from variables and constants, using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents....
in one variable with coefficients in a field.
Z[i]: the ring of Gaussian integers
Z[ω] (where ω is a cube root of 1): the Eisenstein integers

Examples of integral domains that are not PIDs:
Z[x]: the ring of all polynomials with integer coefficients.
It is not principal because the ideal generated by 2 and X is an example of an ideal that cannot be generated by a single polynomial.
K[x,y]: The ideal (x,y) is not principal.

Modules
The key result here is the structure theorem for finitely generated modules over a principal ideal domain
Structure theorem for finitely generated modules over a principal ideal domain

In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules can be uniquely decomposed in much the same way that integers have a prime f...
. This yields that if R is a principal ideal domain, and M is a finitely generated R-module, then a minimal generating set for M has properties somewhat akin to those of a basis for a finite-dimensional vector space over a field. This is something of an over-simplification, since there can be nonzero elements r, m of R and M respectively such that r.m = 0, unlike the case of a vector-space over a field, and this necessitates a more complicated statement.

If M is a free module over a principal ideal domain R, then every submodule of M is again free. This does not hold for modules over arbitrary rings, as the example of modules over shows.

Properties
In a principal ideal domain, any two elements a,b have a greatest common divisor
Greatest common divisor

In mathematics, the greatest common divisor , sometimes known as the greatest common factor or highest common factor , of two non-zero integers, is the largest positive integer that divisor both numbers without remainder....
, which may be obtained as a generator of the ideal (a,b).

All Euclidean domains
Euclidean domain

In abstract algebra, a Euclidean domain is a type of Ring in which the Euclidean algorithm applies. Euclidean domains possess many important properties similar to the integers: for example, the fundamental theorem of arithmetic holds in any Euclidean domain....
are principal ideal domains, but the converse is not true. An example of a principal ideal domain that is not a Euclidean domain is the ring

Every principal ideal domain is a unique factorization domain
Unique factorization domain

In mathematics, a unique factorization domain is, roughly speaking, a commutative ring in which every element, with special exceptions, can be uniquely written as a product of prime elements, analogous to the fundamental theorem of arithmetic for the integers....
(UFD). The converse does not hold since for any field K, K[X,Y] is a UFD but is not a PID (to prove this look at the ideal generated by It is not the whole ring since it contains no polynomials of degree 0, but it cannot be generated by any one single element).

Every principal ideal domain is Noetherian
Noetherian ring

In abstract algebra, a Noetherian ring, named after Emmy Noether, is a ring that satisfies the ascending chain condition on ideal . Explicitly this means: given an increasing sequence of ideals...
.
In all unital rings, maximal ideal
Maximal ideal

In mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal amongst all proper ideals, i.e. which is not contained in any other proper ideal of the ring ....
s are prime
Prime ideal

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. This article only covers ideals of ring theory....
. In principal ideal domains a near converse holds: every nonzero prime ideal is maximal.
All principal ideal domains are integrally closed
Integrally closed

In mathematics, more specifically in abstract algebra, the concept of integrally closed has two meanings, one for group and one for ring ....
.

The previous three statements give the definition of a Dedekind domain
Dedekind domain

In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product...
, and hence every principal ideal domain is a Dedekind domain.

So that PID ⊆ Dedekind∩UFD . However there is another theorem which states that any unique factorisation domain that is a Dedekind domain is also a principal ideal domain. Thus we get the reverse inclusion Dedekind∩UFD ⊆ PID, and then this shows equality and hence, Dedekind∩UFD = PID. (Note that condition (3) above is redundant in this equality, since all UFDs are integrally closed.)