Principal ideal
Encyclopedia
In ring theory
Ring theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...

, a branch of abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

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

 I in a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

 R that is generated by a single element a of R.

More specifically:
  • a left principal ideal of R is a subset
    Subset
    In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

     of R of the form Ra := {ra : r in R};
  • a right principal ideal is a subset of the form aR := {ar : r in R};
  • a two-sided principal ideal is a subset of the form RaR := {r1as1 + ... + rnasn : r1,s1,...,rn,sn in R}.

If R is a 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....

, then the above three notions are all the same.
In that case, it is common to write the ideal generated by a as ⟨a⟩.

Not all ideals are principal.
For example, consider the commutative ring C[x,y] of all polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

s in two variables
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...

 x and y, with complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

 coefficients.
The ideal ⟨x,y⟩ generated by x and y, which consists of all the polynomials in C[x,y] that have zero
0 (number)
0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...

 for the constant term
Constant term
In mathematics, a constant term is a term in an algebraic expression has a value that is constant or cannot change, because it does not contain any modifiable variables. For example, in the quadratic polynomialx^2 + 2x + 3,\ the 3 is a constant term....

, is not principal.
To see this, suppose that p were a generator for ⟨x,y⟩; then x and y would both be divisible by p, which is impossible unless p is a nonzero constant.
But zero is the only constant in ⟨x,y⟩, so we have a contradiction
Contradiction
In classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other...

.

A ring in which every ideal is principal is called principal, or a principal ideal ring
Principal ideal ring
In mathematics, a principal right ideal ring is a ring R in which every right ideal is of the form xR for some element x of R...

.
A 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, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...

 (PID) is an integral domain that is principal.
Any PID must be 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...

; the normal proof of unique factorization in the integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s (the so-called fundamental theorem of arithmetic
Fundamental theorem of arithmetic
In 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 in any PID.

Also, any Euclidean domain
Euclidean domain
In mathematics, more specifically in abstract algebra and ring theory, a Euclidean domain is a ring that can be endowed with a certain structure – namely a Euclidean function, to be described in detail below – which allows a suitable generalization of the Euclidean algorithm...

 is a PID; the algorithm used to calculate greatest common divisor
Greatest common divisor
In mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...

s may be used to find a generator of any ideal.
More generally, any two principal ideals in a commutative ring have a greatest common divisor in the sense of ideal multiplication.
In principal ideal domains, this allows us to calculate greatest common divisors of elements of the ring, up to multiplication by a unit
Unit (ring theory)
In mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that...

; we define gcd(a,b) to be any generator of the ideal ⟨a,b⟩.

For 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 of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors...

 R, we may also ask, given a non-principal ideal I of R, whether there is some extension S of R such that the ideal of S generated by I is principal (said more loosely, I becomes principal in S).
This question arose in connection with the study of rings of algebraic integer
Algebraic integer
In number theory, an algebraic integer is a complex number that is a root of some monic polynomial with coefficients in . The set of all algebraic integers is closed under addition and multiplication and therefore is a subring of complex numbers denoted by A...

s (which are examples of Dedekind domains) in number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, and led to the development of class field theory
Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...

 by Teiji Takagi
Teiji Takagi
Teiji Takagi was a Japanese mathematician, best known for proving the Takagi existence theorem in class field theory....

, Emil Artin
Emil Artin
Emil Artin was an Austrian-American mathematician of Armenian descent.-Parents:Emil Artin was born in Vienna to parents Emma Maria, née Laura , a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of Armenian descent...

, David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

, and many others.

The principal ideal theorem of class field theory
Principal ideal theorem
In mathematics, the principal ideal theorem of class field theory, a branch of algebraic number theory, is the statement that for any algebraic number field K and any ideal I of the ring of integers of K, if L is the Hilbert class field of K, thenIO_L\ is a principal ideal αOL, for OL the ring of...

 states that every integer ring R (i.e. the ring of integers
Ring of integers
In mathematics, the ring of integers is the set of integers making an algebraic structure Z with the operations of integer addition, negation, and multiplication...

 of some number field) is contained in a larger integer ring S which has the property that every ideal of R becomes a principal ideal of S.
In this theorem we may take S to be the ring of integers of the Hilbert class field
Hilbert class field
In algebraic number theory, the Hilbert class field E of a number field K is the maximal abelian unramified extension of K. Its degree over K equals the class number of K and the Galois group of E over K is canonically isomorphic to the ideal class group of K using Frobenius elements for prime...

 of R; that is, the maximal unramified
Ramification
In mathematics, ramification is a geometric term used for 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign...

 abelian extension (that is, Galois extension
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...

 whose Galois group is abelian
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

) of the fraction field of R, and this is uniquely determined by R.

Krull's principal ideal theorem
Krull's principal ideal theorem
In commutative algebra, Krull's principal ideal theorem, named after Wolfgang Krull , gives a bound on the height of a principal ideal in a Noetherian ring...

 states that if R is a Noetherian ring and I is a principal, proper ideal of R, then I has height at most one.
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