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.

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 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

Two ideals A and B in the commutative ring R are called coprime ( or comaximal ) if A + B = R. This generalizes Bézout's identity: with this definition, two principal ideals ( a ) and ( b ) in the ring of integers Z are coprime if and only if a and b are coprime.

Its most common meaning, however, pertains to two principal ideals ; that social life should be structured through well-defined and well-regulated classes ( varna ), and that an individual's life within a class should be organized into defined stages ( ashrama, see dharmasastra ).

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 ).

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.

Prime ideals, which generalize prime elements in the sense that the principal ideal generated by a prime element is a prime ideal, are an important tool and object of study in commutative algebra, algebraic number theory and algebraic geometry.

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.

# Every finitely generated ideal of A is principal ( i. e., A is a Bézout domain ) and A satisfies the ascending chain condition on principal ideals.

* 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.

** Bézout domain, an integral domain in which the sum of two principal ideals is again a principal ideal

* In the ring Z of integers the maximal ideals are the principal ideals generated by a prime number.

* More generally, all nonzero prime ideals are maximal in a principal ideal domain.

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.

In the summary of the principal events of the campaign compiled from the official records there are only ten days which show no fighting.

SBA works closely with the principal property disposal installations of the Federal Government in reviewing proposed sales programs and identifying those types of property that small business concerns are most likely to be interested in purchasing.

If the less developed countries are to be persuaded to adopt a long-term approach, the United States, as the principal supplier of external aid, must be prepared to give long-term commitments.

after completing the payments prescribed by paragraphs ( 2 ) and ( 3 ) of this subsection, to make payments, from time to time in ratable proportions, on account of the unpaid principal of all awards in the principal amount of more than $1,000, according to the proportions which the unpaid principal of such awards bear to the total amount in the fund available for distribution at the time such payments are made ; ;

Their locations in all parts of the United States, and their locations in the several kinds of educational and research institutions that are the principal homes of our intellectual and artistic strengths also are factors in the Trustees' minds.

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 ; ;

During summer vacation periods these records are stored in the office of the principal.

Of the various particle accelerators, the Van De Graff machines, resonant transformers, and linear accelerators are the principal ones available for commercial use.

In other countries where cereal grains are not among the principal crops of a region, starchy tubers or roots are processed for starch.

These, he said, are `` two of the principal underlying causes for family breakups leading to ADC ''.

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.

There are three principal families of abugidas, depending on whether vowels are indicated by modifying consonants by diacritics, distortion, or orientation.

They are derived from the characteristics of their spectroscopic lines: sharp, principal, diffuse, and fundamental, the rest being named in alphabetical order ( omitting j ).

The principal abalone farming regions are China, Taiwan, Japan, and Korea.

His principal theological works are a commentary in three volumes on the Books of the Sentences of Peter Lombard ( Magister Sententiarum ), and the Summa Theologiae in two volumes.

His principal sources are the Physics ( book 7 ), Metaphysics ( book 12 ), and the Pseudo-Aristotelian On the Universe.

However we are told by Longinus that Ammonius wrote nothing, and if Ammonius was the principal influence on Plotinus, then it is unlikely that Ammonius would have been a Christian.

Aldosterone's effects are on the distal convoluted tubule and collecting duct of the kidney where it causes increased reabsorption of sodium and increased excretion of both potassium ( by principal cells ) and hydrogen ions ( by intercalated cells of the collecting duct ).

Mycenae and Tiryns are the two principal sites on which evidence of a prehistoric civilization was remarked long ago by the classical Greeks.

There are difficulties in this story, of which the following are the principal elements: