We denote the subgroup of principal fractional ideals by Prin ( R ).

A domain R is a PID if and only if every fractional ideal is principal.

, since two principal fractional ideals and are equal iff is a unit in R.

For a general domain R, it is meaningful to take the quotient of the monoid Frac ( R ) of all fractional ideals by the submonoid Prin ( R ) of principal fractional ideals.

We note that for an arbitrary domain one may define the Picard group Pic ( R ) as the group of invertible fractional ideals Inv ( R ) modulo the subgroup of principal fractional ideals.

In particular, it asserts that all fractional ideals are principal, a statement which is false whenever is not a PID.

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 >.

where I < sub > K </ sub > is the group of fractional ideals of K, and P < sub > K </ sub > is the group of principal fractional ideals of K, that is, ideals of the form aO < sub > K </ sub > where a is a unit of K.

where now P < sub > K </ sub >< sup >+</ sup > is the group of totally positive principal fractional ideals of K ; that is, ideals of the form aO < sub > K </ sub > where a is a unit of K such that σ ( a ) is positive for every embedding

* 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 principal defender of this view of primary experience as `` causal efficacy '' is Alfred North Whitehead.

but his principal theme is that the intrigues of the Tories, `` our Popish or Jacobite Party '', pose an immediate threat to Church and State.

This is the principal point made in this final section of Englishman No. 57, and it caps Steele's efforts in his other writing of these months to counteract the notion of the Tories as a `` Church Party '' supported by the body of the clergy.

But when the situation was so complicated that even Nogaret, one of the principal actors in the drama, could misinterpret the pope's motives, it is possible that Othon and his companions, equally baffled, attributed their difficulties to a more immediate cause.

We submit that this is a most desirable effect of the law -- and one of its principal aims.

A primary function is the operation of a Government Bid Center, which receives bids daily from the Federal Government's principal purchasing agencies.

Subject to the limitations hereinafter provided, the Secretary of the Treasury is authorized and directed to pay, as prescribed by Section 8 of this Title, an amount not exceeding the principal of each award, plus accrued interests on such awards as bear interest, certified pursuant to Section 5 of this Title, in accordance with the award.

The Strategic Air Command is the principal element of our long-range nuclear capability.

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..

But in such an important question, we would be satisfied if the judgment were that the principal objection to the identity of forces which produce electricity and magnetism were only a difficulty, and not a thing which is contrary to it.

There are three principal feed bunk types for dairy and beef cattle: ( 1 ) Fence-line bunks -- cattle eat from one side while feed is put in from the opposite side of the fence by self-unloading wagons ; ;

This is a pilot operation sponsored by a new entity chartered in Delaware as the Tri-State Pipeline Corporation, with principal offices in New York State.

Miller ( '50 ) is the principal antagonist of this viewpoint.

Emotional maturity is the result of many factors, the principal ones being the experiences of the first few years of the child's life.

Perhaps one way to sharpen our sense of the modernity of Utopian communism is to contrast it with the principal earlier types of communistic theory.

-- One of the principal aims of anionic polymerization techniques is the synthesis of polymers of extremely narrow molecular weight distribution.

If it is owned, taxes must be paid, and if the place is not free of mortgage, there will be interest and payments on the principal to take care of.

The role of an earthquake in starting the destruction of whole cities is tremendously frightening, but fire may actually be the principal agent in a particular disaster.

In general, friendly contact with a member followed by contact with a clergyman will account for a major share of recruitment by the churches, making it quite evident that the extension of economic integration through co-optation is the principal form of mission in the contemporary church ; ;

The new school superintendent is Harry Davis, a veteran agriculture teacher, who defeated Felix Bush, a school principal and chairman of the Miller County Democratic Executive Committee.

The principal of the school announced that -- despite the help of private tutors in Hollywood and Philadelphia -- Fabian is a 10-o'clock scholar in English and mathematics.

While it must be said that these same Protestants have built some new churches during this period, and that religious population shifts have emptied churches, a principal reason for this phenomenon of redundancy is that fewer Protestants are going to church.

However, my principal objection in this sort of novel is to the hackneyed treatment of race-drivers, pilots, submariners, atomic researchers, and all the machine-masters of our age as brooding mystics or hysterical fatalists.