Editing Principal ideal

{{Short description|Ring ideal generated by a single element of the ring}}
{{Multiple issues|
{{More citations needed|date=October 2013}}
{{No footnotes|date=October 2013}}
}}

In [[mathematics]], specifically [[ring theory]], a '''principal ideal''' is an [[ideal (ring theory)|ideal]] <math>I</math> in a [[ring (mathematics)|ring]] <math>R</math> that is generated by a single element <math>a</math> of <math>R</math> through multiplication by every element of <math>R.</math> The term also has another, similar meaning in [[order theory]], where it refers to an [[ideal (order theory)|(order) ideal]] in a [[poset]] <math>P</math> generated by a single element <math>x \in P,</math> which is to say the set of all elements less than or equal to <math>x</math> in <math>P.</math>

The remainder of this article addresses the ring-theoretic concept.

==Definitions==
* A ''left principal ideal'' of <math>R</math> is a [[subset]] of <math>R</math> given by <math>Ra = \{ra : r \in R\}</math> for some element <math>a.</math>
* A ''right principal ideal'' of <math>R</math> is a subset of <math>R</math> given by <math>aR = \{ar : r \in R\}</math> for some element <math>a.</math>
* A ''two-sided principal ideal'' of <math>R</math> is a subset of <math>R</math> given by <math>RaR = \{r_1 a s_1 + \ldots + r_n a s_n: r_1,s_1, \ldots, r_n, s_n \in R\}</math> for some element <math>a,</math> namely, the set of all finite sums of elements of the form <math>ras.</math>

While the definition for two-sided principal ideal may seem more complicated than for the one-sided principal ideals, it is necessary to ensure that the ideal remains closed under addition.{{r|n=df3ed|pp=251-252|r={{cite book|last=Dummit|first=David S.|last2=Foote|first2=Richard M.|title=Abstract Algebra|edition=3rd|publisher=John Wiley & Sons|publication-place=New York|date=2003-07-14|isbn=0-471-43334-9}}}}

If <math>R</math> is a [[commutative ring]], then the above three notions are all the same. In that case, it is common to write the ideal generated by <math>a</math> as <math>\langle a \rangle</math> or <math>(a).</math>

==Examples and non-examples==
* The principal ideals in the (commutative) ring <math>\mathbb{Z}</math> are <math>\langle n \rangle = n\mathbb{Z}=\{\ldots,-2n,-n,0,n,2n,\ldots\}.</math> In fact, every ideal of <math>\mathbb{Z}</math> is principal (see {{section link|#Related definitions}}).
* In any ring <math>R</math>, the sets <math>\{0\}= \langle 0\rangle</math> and <math>R=\langle 1\rangle</math> are principal ideals.
* For any ring <math>R</math> and element <math>a,</math> the ideals <math>Ra,aR,</math> and <math>RaR</math> are respectively left, right, and two-sided principal ideals, by definition. For example, <math>\langle \sqrt{-3} \rangle</math> is a principal ideal of <math>\mathbb{Z}[\sqrt{-3}].</math>
* In the commutative ring <math>\mathbb{C}[x,y]</math> of [[complex number|complex]] [[polynomials]] in two [[variable (mathematics)|variable]]s, the set of polynomials that vanish everywhere on the set of points <math>\{(x,y)\in\mathbb{C}^2\mid x=0\}</math> is a principal ideal because it can be written as <math>\langle x\rangle</math> (the set of polynomials divisible by <math>x</math>).
* In the same ring <math>\mathbb{C}[x,y]</math>, the ideal <math>\langle x, y\rangle</math> generated by both <math>x</math> and <math>y</math> is ''not'' principal. (The ideal <math>\langle x, y\rangle</math> is the set of all polynomials with zero for the [[constant term]].) To see this, suppose there was a generator <math>p</math> for <math>\langle x,y\rangle,</math> so <math>\langle x, y\rangle=\langle p\rangle.</math> Then <math>\langle p\rangle</math> contains both <math>x</math> and <math>y,</math> so <math>p</math> must divide both <math>x</math> and <math>y.</math> Then <math>p</math> must be a nonzero constant polynomial. This is a contradiction since <math>p\in\langle p\rangle</math> but the only constant polynomial in <math>\langle x, y\rangle,</math> is the zero polynomial.
* In the ring <math>\mathbb{Z}[\sqrt{-3}] = \{a + b\sqrt{-3}: a, b\in \mathbb{Z} \},</math> the numbers where <math>a + b</math> is even form a non-principal ideal.  This ideal forms a regular hexagonal lattice in the complex plane.  Consider <math>(a,b) = (2,0)</math> and <math>(1,1).</math>  These numbers are elements of this ideal with the same norm (two), but because the only units in the ring are <math>1</math> and <math>-1,</math> they are not associates.

==Related definitions==
A ring in which every ideal is principal is called ''principal'', or a ''[[principal ideal ring]]''. A ''[[principal ideal domain]]'' (PID) is an [[integral domain]] in which every ideal is principal. Any PID is a [[unique factorization domain]]; the normal proof of unique factorization in the [[integer]]s (the so-called [[fundamental theorem of arithmetic]]) holds in any PID.

As an example, <math>\mathbb{Z}</math> is a principal ideal domain, which can be shown as follows. Suppose <math>I=\langle n_1, n_2, \ldots\rangle</math> where <math>n_1\neq 0,</math> and consider the surjective homomorphisms <math>\mathbb{Z}/\langle n_1\rangle \rightarrow \mathbb{Z}/\langle n_1, n_2\rangle \rightarrow \mathbb{Z}/\langle n_1, n_2, n_3\rangle\rightarrow \cdots.</math> Since <math>\mathbb{Z}/\langle n_1\rangle</math> is finite, for sufficiently large <math>k</math> we have <math>\mathbb{Z}/\langle n_1, n_2, \ldots, n_k\rangle = \mathbb{Z}/\langle n_1, n_2, \ldots, n_{k+1}\rangle = \cdots.</math> Thus <math>I=\langle n_1, n_2, \ldots, n_k\rangle,</math> which implies <math>I</math> is always finitely generated.  Since the ideal <math>\langle a,b\rangle</math> generated by any integers <math>a</math> and <math>b</math> is exactly <math>\langle \mathop{\mathrm{gcd}}(a,b)\rangle,</math> by induction on the number of generators it follows that <math>I</math> is principal.

==Properties==
Any [[Euclidean domain]] is a [[Principal ideal domain|PID]]; the algorithm used to calculate [[greatest common divisor]]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 (ring theory)|unit]]; we define <math>\gcd(a, b)</math> to be any generator of the ideal <math>\langle a, b \rangle.</math>

For a [[Dedekind domain]] <math>R,</math> we may also ask, given a non-principal ideal <math>I</math> of <math>R,</math> whether there is some extension <math>S</math> of <math>R</math> such that the ideal of <math>S</math> generated by <math>I</math> is principal (said more loosely, <math>I</math> ''becomes principal'' in <math>S</math>).
This question arose in connection with the study of rings of [[algebraic integer]]s (which are examples of Dedekind domains) in [[number theory]], and led to the development of [[class field theory]] by [[Teiji Takagi]], [[Emil Artin]], [[David Hilbert]], and many others.

The [[principal ideal theorem|principal ideal theorem of class field theory]] states that every integer ring <math>R</math> (i.e. the [[ring of integers]] of some [[number field]]) is contained in a larger integer ring <math>S</math> which has the property that ''every'' ideal of <math>R</math> becomes a principal ideal of <math>S.</math> In this theorem we may take <math>S</math> to be the ring of integers of the [[Hilbert class field]] of <math>R</math>; that is, the maximal [[Ramification (mathematics)|unramified]] abelian extension (that is, [[Galois extension]] whose [[Galois group]] is [[abelian group|abelian]]) of the fraction field of <math>R,</math> and this is uniquely determined by <math>R.</math>

[[Krull's principal ideal theorem]] states that if <math>R</math> is a Noetherian ring and <math>I</math> is a principal, proper ideal of <math>R,</math> then <math>I</math> has [[height (ring theory)|height]] at most one.

== See also ==
* [[Ascending chain condition for principal ideals]]

==References==
{{reflist}}
* {{cite book
| last =Gallian |first = Joseph A.
| date = 2017
| edition = 9th
| title = Contemporary Abstract Algebra
| publisher = Cengage Learning
| isbn = 978-1-305-65796-0
}}

[[Category:Ideals (ring theory)]]
[[Category:Commutative algebra]]