In

p.170

Proposition 7.3.3. Any

Principal ring

on

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, a principal ideal domain, or PID, is an integral domain
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

in which every ideal
Ideal may refer to:
Philosophy
* Ideal (ethics)
An ideal is a principle
A principle is a proposition or value that is a guide for behavior or evaluation. In law
Law is a system
A system is a group of Interaction, interacting ...

is principal
Principal may refer to:
Title or rank
* Principal (academia)
The principal is the chief executive and the chief academic officer of a university
A university ( la, universitas, 'a whole') is an educational institution, institution of higher ...

, i.e., can be generated by a single element. More generally, a principal ideal ringIn mathematics, a principal right (left) ideal ring is a ring ''R'' in which every right (left) ideal is of the form ''xR'' (''Rx'') for some element ''x'' of ''R''. (The right and left ideals of this form, generated by one element, are called princ ...

is a nonzero commutative ring whose ideals are principal, although some authors (e.g., Bourbaki) refer to PIDs as principal rings. The distinction is that a principal ideal ring may have zero divisor
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

s whereas a principal ideal domain cannot.
Principal ideal domains are thus mathematical objects that behave somewhat like the integer
An integer (from the Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to ...

s, with respect to divisibility
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

: any element of a PID has a unique decomposition into prime element
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s (so an analogue of the fundamental theorem of arithmetic
In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "wh ...

holds); any two elements of a PID have a greatest common divisor
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

(although it may not be possible to find it using the Euclidean algorithm
In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an effi ...

). If and are elements of a PID without common divisors, then every element of the PID can be written in the form .
Principal ideal domains are noetherianIn mathematics, the adjective
In linguistics, an adjective (list of glossing abbreviations, abbreviated ) is a word that grammatical modifier, modifies a noun or noun phrase or describes its referent. Its Semantics, semantic role is to change inf ...

, they are integrally closed, they are unique factorization domain
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s and Dedekind domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathema ...

s. All Euclidean domain
In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a #Definition, Euclidean function which allows a suitable generalization of the Euclidean division of ...

s and all fields
File:A NASA Delta IV Heavy rocket launches the Parker Solar Probe (29097299447).jpg, FIELDS heads into space in August 2018 as part of the ''Parker Solar Probe''
FIELDS is a science instrument on the ''Parker Solar Probe'' (PSP), designed to mea ...

are principal ideal domains.
Principal ideal domains appear in the following chain of class inclusions:
Examples

Examples include: * $K$: anyfield
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

,
* $\backslash mathbb$: the ring of integer
An integer (from the Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to ...

s,
* $K;\; href="/html/ALL/s/.html"\; ;"title="">$Gaussian integers
In number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number t ...

,
* $\backslash mathbb$omega
Omega (; capital
Capital most commonly refers to:
* Capital letter
Letter case (or just case) is the distinction between the letters that are in larger uppercase or capitals (or more formally ''majuscule'') and smaller lowercase (o ...

/math> (where $\backslash omega$ is a primitive cube root of 1): the Eisenstein integers
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

,
* Any discrete valuation ringIn abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra ...

, for instance the ring of -adic integers $\backslash mathbb\_p$.
Non-examples

Examples of integral domains that are not PIDs: * $\backslash mathbb$sqrt
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

/math> is an example of a ring which is not a unique factorization domain
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, since $4\; =\; 2\backslash cdot\; 2\; =\; (1+\backslash sqrt)(1-\backslash sqrt).$ Hence it is not a principal ideal domain because principal ideal domains are unique factorization domains.
*$\backslash mathbb;\; href="/html/ALL/s/.html"\; ;"title="">$, y
The comma is a punctuation
Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...

/math>: rings of polynomials in two variables. The ideal $\backslash langle\; x,\; y\; \backslash rangle$ is not principal.
*Most rings of algebraic integers are not principal ideal domains because they have ideals which are not generated by a single element. This is one of the main motivations behind Dedekind's definition of Dedekind domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathema ...

s since a prime integer can no longer be factored into elements, instead they are prime ideals. In fact many $\backslash mathbb;\; href="/html/ALL/s/zeta\_p.html"\; ;"title="zeta\_p">zeta\_p$Modules

The key result is the structure theorem: If ''R'' is a principal ideal domain, and ''M'' is a finitely generated ''R''-module, then $M$ is a direct sum of cyclic modules, i.e., modules with one generator. The cyclic modules are isomorphic to $R/xR$ for some $x\backslash in\; R$ (notice that $x$ may be equal to $0$, in which case $R/xR$ is $R$). If ''M'' is afree module
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

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 $(2,X)\; \backslash subseteq\; \backslash mathbb;\; href="/html/ALL/s/.html"\; ;"title="">$Properties

In a principal ideal domain, any two elements have agreatest common divisor
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

, which may be obtained as a generator of the ideal .
All Euclidean domain
In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a #Definition, Euclidean function which allows a suitable generalization of the Euclidean division of ...

s 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 $\backslash mathbb\backslash left;\; href="/html/ALL/s/frac\_2\backslash right.html"\; ;"title="frac\; 2\backslash right">frac\; 2\backslash right$ In this domain no and exist, with , so that $(1+\backslash sqrt)=(4)q+r$, despite $1+\backslash sqrt$ and $4$ having a greatest common divisor of .
Every principal ideal domain is a unique factorization domain
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

(UFD). The converse does not hold since for any UFD , the ring of polynomials in 2 variables is a UFD but is not a PID. (To prove this look at the ideal generated by $\backslash left\backslash langle\; X,Y\; \backslash right\backslash rangle.$ 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 NoetherianIn mathematics, the adjective
In linguistics, an adjective (list of glossing abbreviations, abbreviated ) is a word that grammatical modifier, modifies a noun or noun phrase or describes its referent. Its Semantics, semantic role is to change inf ...

.
#In all unital rings, maximal idealIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

s are prime
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

. In principal ideal domains a near converse holds: every nonzero prime ideal is maximal.
#All principal ideal domains are integrally closed.
The previous three statements give the definition of a Dedekind domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathema ...

, and hence every principal ideal domain is a Dedekind domain.
Let ''A'' be an integral domain. Then the following are equivalent.
# ''A'' is a PID.
# Every prime ideal of ''A'' is principal.
# ''A'' is a Dedekind domain that is a UFD.
# Every finitely generated ideal of ''A'' is principal (i.e., ''A'' is a Bézout domainIn mathematics, a Bézout domain is a form of a Prüfer domain. It is an integral domain in which the sum of two principal ideals is again a principal ideal. This means that for every pair of elements a Bézout identity holds, and that every finitel ...

) and ''A'' satisfies the ascending chain condition on principal ideals.
# ''A'' admits a Dedekind–Hasse norm.Hazewinkel, Gubareni & Kirichenko (2004)p.170

Proposition 7.3.3. Any

Euclidean norm
Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called pa ...

is a Dedekind-Hasse norm; thus, (5) shows that a Euclidean domain is a PID. (4) compares to:
* An integral domain is a UFD if and only if it is a GCD domain
In mathematics, a GCD domain is an integral domain ''R'' with the property that any two elements have a greatest common divisor
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number th ...

(i.e., a domain where every two elements have a greatest common divisor) satisfying the ascending chain condition on principal ideals.
An integral domain is a Bézout domainIn mathematics, a Bézout domain is a form of a Prüfer domain. It is an integral domain in which the sum of two principal ideals is again a principal ideal. This means that for every pair of elements a Bézout identity holds, and that every finitel ...

if and only if any two elements in it have a gcd ''that is a linear combination of the two.'' A Bézout domain is thus a GCD domain, and (4) gives yet another proof that a PID is a UFD.
See also

*Bézout's identity
In elementary number theory, Bézout's identity (also called Bézout's lemma) is the following theorem:
Here the greatest common divisor of 0 and 0 is taken to be 0. The integers ''x'' and ''y'' are called Bézout coefficients for (''a'', ''b''); ...

Notes

References

*Michiel Hazewinkel
Michiel Hazewinkel (born 22 June 1943) is a Dutch mathematician, and Emeritus Professor of Mathematics at the Centrum Wiskunde & Informatica, Centre for Mathematics and Computer Science and the University of Amsterdam, particularly known for his 1 ...

, Nadiya Gubareni, V. V. Kirichenko. ''Algebras, rings and modules''. Kluwer Academic Publishers
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 i ...

, 2004.
* John B. Fraleigh, Victor J. Katz. ''A first course in abstract algebra''. Addison-Wesley Publishing Company. 5 ed., 1967.
* Nathan Jacobson
Nathan Jacobson (October 5, 1910 – December 5, 1999) was an American
American(s) may refer to:
* American, something of, from, or related to the United States of America, commonly known as the United States
The United States of America ( ...

. Basic Algebra I. Dover, 2009.
* Paulo Ribenboim. ''Classical theory of algebraic numbers''. Springer, 2001.
External links

Principal ring

on

MathWorld
''MathWorld'' is an online mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geome ...

{{DEFAULTSORT:Principal Ideal Domain
Commutative algebra
Ring theory