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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.
* 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.

principal and domain
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 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.
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 ).
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 ).

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.
In principal ideal domains a near converse holds: every nonzero prime ideal is maximal.
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.
** Principal ideal domain, an integral domain in which every ideal is principal
As for every principal ideal domain, the Gaussian integers form also a unique factorization domain.
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.
* More generally, all nonzero prime ideals are maximal in a principal ideal domain.
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.
* If I is a nonzero ideal of a Dedekind domain A, then is a principal Artinian ring.
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
Alexander proved that the Alexander ideal is nonzero and always principal.

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