* 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.

* R is a principal ideal domain.

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.

Important examples are Euclidean domains and principal ideal domains.

: 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.

PL / I's principal domains are data processing, numerical computation, scientific computing, and systems programming ; it supports recursion, structured programming, linked data structure handling, fixed-point, floating-point, complex, character string handling, and bit string handling.

All Euclidean domains and all fields are principal ideal domains.

: Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields

More, the U.S. action was hailed by a principal opposition leader, Dr. Juan Bosch, as having saved `` many lives and many troubles in the near future ''.

On the middle Mississippi a principal post was to be located near the mouth of the Arkansas.

Another was unveiled at St Mildred's Church, Bread St, London, in 1932 ; that church was destroyed in the London Blitz in 1940, but the principal elements of the monument were re-erected in St Mary-le-Bow at the west end of Watling Street, near Saint Paul's Cathedral, in 1968.

The R-99 fleet is one of the principal components of the system, the aircraft are based at Anápolis AB near Brasília and fly 24 hours a day over the Amazonian region.

Among his principal miracles are: ( 1 ) procuring of food for a sick monk and curing the wife of his benefactor ; ( 2 ) escape from hurt when surrounded by wolves ; ( 3 ) obedience of a bear which evacuated a cave at his biddings ; ( 4 ) producing a spring of water near his cave ; ( 5 ) repletion of the Luxeuil granary when empty ; ( 6 ) multiplication of bread and beer for his community ; ( 7 ) curing of the sick monks, who rose from their beds at his request to reap the harvest ; ( 8 ) giving sight to a blind man at Orleans ; ( 9 ) taming a bear, and yoking it to a plough.

Mali has no seaports because it is landlocked, but Koulikoro on the Niger River near Bamako, serves as a principal river port.

Low passes are lacking, and the principal roads and the railroads between France and Spain run only in the lowlands at the western and eastern ends of the Pyrenees, near sea level.

Arguably the principal Isotope Separator On-Line ( ISOL ) is ISOLDE at CERN, which is a joint European facility spread across the Franco-Swiss border near the city of Geneva.

About the same time he published Passavantius,, a satire directed against Pierre Lizet, the former president of the Parliament of Paris, and principal originator of the " fiery chamber " ( chambre ardente ), who, at the time ( 1551 ) was abbot of St. Victor near Paris and publishing a number of polemical writings.

Having succeeded Anderson at Louisville, Sherman now had principal military responsibility for Kentucky, a border state in which Confederate troops held Columbus and Bowling Green and were present near the Cumberland Gap.

Located near the border of California and Nevada, in the Great Basin, east of the Sierra Nevada mountains, Death Valley constitutes much of Death Valley National Park and is the principal feature of the Mojave and Colorado Deserts Biosphere Reserve.

Tissaphernes, who once again had recourse to subtle diplomacy, was beaten by Agesilaus II on the Pactolus near Sardis in 395 BC ; and at last the king yielded to the representations of Pharnabazus, strongly supported by the chiliarch ( vizier ) Tithraustes and by the queen-mother Parysatis, who hated Tissaphernes as the principal cause of the death of her favourite son Cyrus.

In 1776, Chief Netawatwes donated land for another settlement, Lichtenau ( meadow of light ), near present-day Coshocton, then the principal Lenape village in the region.

The Coast Miwok civilization thrived in the Cotati area since at least 2000 BC, with principal villages built near major streams.

Located near the confluence of the West Branch Susquehanna River and Bald Eagle Creek, it is the principal city of the Lock Haven, Pennsylvania, micropolitan statistical area, itself part of the Williamsport – Lock Haven combined statistical area.

Edward S. Marcus High School, located in Flower Mound but near Highland Village, is the city's principal high school.

As early as the year 1772, settlers began locating their lands near where Clarksburg now stands, and in 1773 Daniel Davisson took up, upon which the principal part of the town is now located.

Waupun lies near the edge of the Horicon Marsh, the largest cattail marsh in the United States, and is the principal access point to wildlife viewing for the Horicon Wildlife Refuges.

The principal settlements on its shores are Dingwall near the head, Cromarty near the mouth ( from which the Firth takes its name ), and Invergordon in between.

His principal antagonist was a Seisyll ap Dyfnwal, of Castell Arnallt near Llanover in the valley of the River Usk near Abergavenny, whom he blamed for the death of his uncle Henry.

Deformation of the lithosphere in the plane of the earth ( i. e. such that faults are vertical ) occurs as a result of near horizontal maximum and minimum principal stresses.

Adults are often found near plants, their principal food source being nectar and pollen.

All Euclidean domains are principal ideal domains, but the converse is not true.

The principal seven divisions are each populated by a single adult slave at Manderlay, who congregate daily and converse on a " parade ground ," with roman numerals of the numbers 1 through 7 designating where each slave stands.

According to the minutes of the meeting on April 6, 1840, at which Hyde was dispatched he was " to visit the cities of London, Amsterdam, Constantinople, and Jerusalem ; and also other places that he may deem expedient ; and converse with the priests, rulers, and elders of the Jews, and obtain from them all the information possible, and communicate the same to some principal paper for publication, that it may have a general circulation throughout the United States.