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In a principal ideal domain every nonzero prime ideal is maximal, but this is not true in general.

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## Some Related Sentences

principal and ideal

* Every unital real Banach algebra with no zero divisors

**,**and**in**which**every**__principal____ideal__**is**closed**,****is**isomorphic to the reals**,**the complexes**,**or the quaternions**.**
Bézout's lemma

**is****true****in**any__principal____ideal__**domain****,****but**there are integral domains**in**which it**is****not****true****.**
As noted

**in**the introduction**,**Bézout's identity works**not**only**in**the ring of integers**,****but**also**in**any other__principal____ideal__**domain**( PID ).
then there are elements x and y

**in**R such that ax + by = d**.**The reason: the__ideal__Ra + Rb**is**__principal__and indeed**is**equal to Rd**.**
Also

**every**__ideal__**in****a**Euclidean**domain****is**__principal__**,**which implies**a**suitable generalization of the Fundamental Theorem of Arithmetic:**every**Euclidean**domain****is****a**unique factorization**domain****.**
It

**is**important to compare the class of Euclidean domains with the larger class of__principal____ideal__domains ( PIDs ).
: Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃

__principal____ideal__domains ⊃ Euclidean domains ⊃ fields**In**words

**,**one may define f (

**a**) to be the minimum value attained by g on the set of all non-zero elements of the

__principal__

__ideal__generated by

**a**

**.**

The property ( EF1 ) can be restated as follows: for any

__principal____ideal__I of R with**nonzero**generator b**,**all**nonzero**classes of the quotient ring R / I have**a**representative r with**.****In**modern mathematical language

**,**the

__ideal__generated by

**a**and b

**is**the

__ideal__generated by g alone ( an

__ideal__generated by

**a**single element

**is**called

**a**

__principal__

__ideal__

**,**and all ideals of the integers are

__principal__ideals ).

: Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃

__principal____ideal__domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields**.**
: Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃

__principal____ideal__domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields**.****In**

**a**ring all of whose ideals are

__principal__(

**a**

__principal__

__ideal__

**domain**or PID ),

**this**

__ideal__will be identical with the set of multiples of some ring element d ; then

**this**d

**is**

**a**greatest common divisor of

**a**and b

**.**But the

__ideal__(

**a**

**,**b ) can be useful even when there

**is**no greatest common divisor of

**a**and b

**.**( Indeed

**,**Ernst Kummer used

**this**

__ideal__as

**a**replacement for

**a**gcd

**in**his treatment of Fermat's Last Theorem

**,**although he envisioned it as the set of multiples of some hypothetical

**,**or

__ideal__

**,**ring element d

**,**whence the ring-theoretic term

**.**

principal and domain

The Internet maintains two

__principal__namespaces**,**the__domain__name hierarchy and the Internet Protocol ( IP ) address spaces**.****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

**.**

The distinction

**is**that**a**__principal__**ideal**ring may have zero divisors whereas**a**__principal__**ideal**__domain__cannot**.****In**

**a**

__principal__

**ideal**

__domain__

**,**any two elements

**a**

**,**b have

**a**greatest common divisor

**,**which may be obtained as

**a**generator of the

**ideal**(

**a**

**,**b ).

principal and every

Nonetheless

**,**Wilson believed that**,****in**all cases**,**corporations “ should be erected with caution**,**and inspected with care .” The actions of corporations were clearly circumscribed: “ To__every__corporation**a**name must be assigned ; and by that name alone it can perform legal acts .” For non-binding external actions or transactions**,**corporations enjoyed the same latitude as private individuals ;**but**it was with an eye to internal affairs that many saw__principal__advantage**in**incorporation**.****In**most churches of the Anglican Communion

**,**the Eucharist

**is**celebrated

__every__Sunday

**,**having replaced Morning Prayer as the

__principal__service

**.**

The seal's

__principal__figure**is**Minerva-a symbol appropriate for an organization created**in**the midst of the American Revolution and dedicated to the cultivation of__every__art and science to " advance the interest**,**honour**,**dignity**,**and happiness of**a**free**,**independent**,**and virtuous people**.**
For

**,**since there are four zones of the world**in**which we live**,**and four__principal__winds**,**while the church**is**scattered throughout all the world**,**and the “ pillar and ground ” of the church**is**the gospel and the spirit of life ; it**is**fitting that she should have four pillars**,**breathing out immortality on__every__side**,**and vivifying men afresh**.**
The previous three statements give the definition of

**a**Dedekind**domain****,**and hence__every____principal__**ideal****domain****is****a**Dedekind**domain****.**
* An integral

**domain****is****a**UFD if and only if it**is****a**GCD**domain**( i**.**e.,**a****domain**where__every__two elements has**a**greatest common divisor ) satisfying the ascending chain condition on__principal__ideals**.**
Among the integers

**,**the ideals correspond one-for-one with the non-negative integers:**in****this**ring**,**__every__**ideal****is****a**__principal__**ideal**consisting of the multiples of**a**single non-negative number**.**
On receiving

**a**favorable reply from the Holy See**,**Gediminas issued circular letters**,**dated 25 January 1325**,**to the__principal__Hansa towns**,**offering**a**free access into his domains to men of__every__order and profession from nobles and knights to tillers of the soil**.**
Geometrically

**this**means that__every__contour ellipsoid**is**infinitely thin and has zero volume**in**n-dimensional space**,**as at least one of the__principal__axes has length of zero**.**
It

**is****a**fact that the ring**is****a**__principal__**ideal**ring ; that**is****,**for any**ideal**I**in****,**there exists an integer n**in**I such that__every__element of I**is****a**multiple of n**.**Conversely**,**the set of all multiples of an arbitrary integer n**is**necessarily an**ideal****,**and**is**usually denoted by ( n ).**In**fact

**,**it

**is**possible to give

**a**proof that

**is**

**a**Noetherian ring without appealing to its order structure and

**this**proof applies more generally to

__principal__

**ideal**rings ( i

**.**e., rings

**in**which

__every__

**ideal**

**is**generated by

**a**single element ).

Krull's

__principal__**ideal**theorem states that__every____principal__**ideal****in****a**commutative Noetherian ring has height one ; that**is****,**__every____principal__**ideal****is**contained**in****a****prime****ideal**minimal amongst**nonzero****prime**ideals**.**

principal and nonzero

More generally

**,****a**__principal__**ideal**ring**is****a**__nonzero__commutative ring whose ideals are__principal__**,**although some authors ( e**.**g., Bourbaki ) refer to PIDs as__principal__rings**.**
Equivalently

**,**an element**is****prime**if**,**and only if**,**the__principal__**ideal**generated by**is****a**__nonzero__**prime****ideal****.**
A

__principal__fractional**ideal****is**one of the form for some__nonzero__x**in**K**.**Note that each__principal__fractional**ideal****is**invertible**,**the inverse of being simply**.**
Define

**a**map from K < sup >×</ sup > to the set of all__nonzero__fractional ideals of R by sending**every**element to the__principal__( fractional )**ideal**it generates**.**
The

__principal__fractional ideals are those R-submodules of K generated by**a**single__nonzero__element of K**.**A fractional**ideal**I**is**contained**in**R if**,**and only if**,**it**is**an (' integral ')**ideal**of R**.**
The smaller one

**,**P < sub > m </ sub >,**is**the group of__principal__fractional ideals ( u / v ) where u and v are__nonzero__elements of O < sub > K </ sub > which are**prime**to m < sub > f </ sub >, u ≡ v mod m < sub > f </ sub >, and u / v > 0**in**each of the orderings of m < sub >∞</ sub >.
Let A be

**a**__nonzero__m × n matrix over**a**__principal__**ideal****domain**R**.**There exist invertible and-matrices S**,**T so that the product S A T**is**0.395 seconds.