* 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.

The Internet maintains two principal namespaces, the domain name hierarchy and the Internet Protocol ( IP ) address spaces.

* Principal ideal domain, in abstract algebra, an integral domain in which every ideal is principal

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

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.

The key result is the structure theorem: If R is a principal ideal domain, and M is a finitely

If M is a free module over a principal ideal domain R, then every submodule of M is again free.

An example of a principal ideal domain that is not a Euclidean domain is the ring

Every principal ideal domain is a unique factorization domain ( UFD ).

The extent of such interference -- which may be so slight as to be undetectable at any point where either of the stations renders a usable signal, or may be so great as to virtually destroy the service areas of both stations -- depends on many factors, among the principal ones being the distance between the stations, their respective radiated power, and, of particular significance here, the time of day.

If you have neither a legal residence nor a principal place of business in any Internal Revenue district, your return should be filed with the District Director of Internal Revenue, Baltimore 2, Md..

If your principal place of abode for the tax year is outside the United States ( including Alaska and Hawaii ), Puerto Rico, or the Virgin Islands and you have no legal residence or principal place of business in any Internal Revenue district in the United States, you should file your return with the Office of International Operations, Internal Revenue Service, Washington 25, D.C..

This work has not changed the mind of any of the principal authors in the field, however.

But once Milne had, in his own words, " said goodbye to all that in 70, 000 words " ( the approximate length of his four principal children's books ), he had no intention of producing any reworkings lacking in originality, given that one of the sources of inspiration, his son, was growing older.

Most notably, Ajax is not wounded in any of the battles described in the Iliad, and he is the only principal character on either side who does not receive personal assistance from any of the gods who take part in the battles.

More specifically in regards to the Mets, critics point out that with Selig's personal relationship with Wilpon has allowed him to stall any possible removal of Wilpon as that club's principal owner.

The term cardinal at one time applied to any priest permanently assigned or incardinated to a church, or specifically to the senior priest of an important church, based on the Latin cardo ( hinge ), meaning " principal " or " chief ".

His Tàichū (, " Grand Inception ") calendar of 104 BC had a year with the winter solstice in the 11th month and designated as intercalary any calendar month ( a month of 29 or 30 whole days ) during which the sun does not pass a principal term ( that is, remained within the same sign of the zodiac throughout ).

Of the principal actors in the trial, whose lives are recorded after, only Cotton Mather and his ally William Stoughton never admitted any guilt.

The principal advantages of compound turbines are the reduction in size of any one casing, the confinement of the highest pressure to the smaller casing ( which may be made of stronger and more expensive materials ) and the possibility of divided flow in the low-pressure casing for the purpose of equalizing end thrusts.

The ultimate aim of any 1st AD is to ensure the film comes in on schedule while maintaining a working environment in which the Director, principal artists ( Actors ) and crew can be focused on their work.

Francis Xavier reached Japan on 27 July 1549, with Anjiro and three other Jesuits, but he was not permitted to enter any port his ship arrived at until 15 August, when he went ashore at Kagoshima, the principal port of the province of Satsuma on the island of Kyūshū.

This means that any regular Latin verb can be conjugated in any person, number, tense, mood, and voice by knowing which of the four conjugation groups it belongs to, and its principal parts.

The court ruled in Carpenter: " It is well established, as a general proposition, that a person who acquires special knowledge or information by virtue of a confidential or fiduciary relationship with another is not free to exploit that knowledge or information for his own personal benefit but must account to his principal for any profits derived therefrom.

Paul Nagel argues that his political acumen was not any less developed than others were in his day, and notes that Henry Clay, one of the era's most astute politicians, was a principal advisor to Adams and supporter throughout his presidency.

Crawfurd, who opposed Darwinian evolution, " denied any unity to mankind, insisting on immutable, hereditary, and timeless differences in racial character, principal amongst which was the ' very great ' difference in ' intellectual capacity.

Orlando Bridgman, who upon his submission to Cromwell had been permitted to practice the law in a private manner, and under that colour had served both as spy and agent for his master, was entrusted with the principal management of this tragic scene ; and in his charge to the Grand Jury, had the assurance to tell them ' That no authority, no single person, or community of men ; not the people collectively or representatively, had any coercive power over the King of England.

Screw dislocations may slip in the direction of any lattice plane containing the dislocation, while the principal driving force for " dislocation climb " is the movement or diffusion of vacancies through a crystal lattice.