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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.
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Before the Juniors entered the ring the Steward announced that after all Juniors had moved their dogs around the ring and set them up, they could relax with their dogs.
After the judge moved all the dogs individually, she selected several from the group and placed them in the center of the ring.
If a is algebraic over K, then K, the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. In the special case where K = Q is the field of rational numbers, Q is an example of an algebraic number field.
This ring has the property that a · a = a for all a in A ; rings with this property are called Boolean rings.
There were twelve rules in all, and they specified that fights should be " a fair stand-up boxing match " in a 24-foot-square or similar ring.
When there are no enemy discs on the board, many ( but not all ) rules also state that a player must shoot for the centre of the board, and a shot disc must finish either completely inside the 15-point guarded ring line, or ( depending on the specifics of the rules ) be inside or touching this line.
* The spectrum of any commutative ring with the Zariski topology ( that is, the set of all prime ideals ) is compact, but never Hausdorff ( except in trivial cases ).
With the 9 main keys, ( operated by the index, middle, and ring fingers ), 2 prefix keys and one delete key, the EkaPad can produce all the inputs of a standard qwerty keyboard with one, two, and a few three finger chords.
In his acceptance speech, Powell reminded Americans that " It is for America, the Land of the Free and the Home of the Brave, to help freedom ring across the globe, unto all the peoples thereof.
The first inductee was Bob Lilly in 1975 and by 2005, the ring contained 17 names, all former Dallas players except for one head coach and one general manager / president.
For square matrices with entries in a non-commutative ring, for instance the quaternions, there is no unique definition for the determinant, and no definition that has all the usual properties of determinants over commutative rings.
Stated differently, a ring is a division ring if and only if the group of units is the set of all non-zero elements.
When his father's ring was sent to him, he begged that his father would show mercy to his mother, and that all his companions would plead with Henry to set her free.
Some authors also require the domain of the Euclidean function be the entire ring R ; this can always be accommodated by adding 1 to the values at all nonzero elements, and defining the function to be 0 at the zero element of R, but the result is somewhat awkward in the case of K. The definition is sometimes generalized by allowing the Euclidean function to take its values in any well-ordered set ; this weakening does not affect the most important implications of the Euclidean property.
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.
As in the simple cases above, one may enumerate all distinct four-colorings of the ring ; any coloring that can be extended without modification to a coloring of the configuration is called initially good.
Surrounding all is a ring or horseshoe-shaped layer of raw coal, usually kept damp and tightly packed to maintain the shape of the fire's heart and to keep the coal from burning directly so that it " cooks " into coke first.
This paper introduced what is now called the ring of Gaussian integers, the set of all complex numbers a + bi where a and b are integers.
< http :// www. chm. bris. ac. uk / motm / hemoglobin / hemoglobjm. htm >.</ ref > The iron ion, which is the site of oxygen binding, coordinates with the four nitrogens in the center of the ring, which all lie in one plane.
However, this all changed in 450 when Honoria, sister of the Western Roman Emperor Valentinian III, sent Attila a ring and requested his help to escape her betrothal to a senator.
ring and whose
Starting with a ring A, we get a unital associative R-algebra by providing a ring homomorphism whose image lies in the center of A.
A benzodiazepine ( sometimes colloquially " benzo "; often abbreviated " BZD ") is a psychoactive drug whose core chemical structure is the fusion of a benzene ring and a diazepine ring.
Prior, whose authority is great in the origin of popular names, says " It seems probably that the name was in the first place, foxes ' glew, or music, in reference to the favourite instrument of an earlier time, a ring of bells hung on an arched support, the tintinnabulum "... we cannot quite agree with Dr.
Although most often used for matrices whose entries are real or complex numbers, the definition of the determinant only involves addition, subtraction and multiplication, and so it can be defined for square matrices with entries taken from any commutative ring.
In abstract algebra, a field is a commutative ring which contains a multiplicative inverse for every nonzero element, equivalently a ring whose nonzero elements form an abelian group under multiplication.
The macrolides are a group of drugs ( typically antibiotics ) whose activity stems from the presence of a macrolide ring, a large macrocyclic lactone ring to which one or more deoxy sugars, usually cladinose and desosamine, may be attached.
Some prokaryotes, including many archaea and the bacterial order Actinomycetales also share homologs of the 20S proteasome, whereas most bacteria possess heat shock genes hslV and hslU, whose gene products are a multimeric protease arranged in a two-layered ring and an ATPase.
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.
The former appears in the title of the Lesser Key of Solomon, a grimoire whose framing tale is Solomon capturing demons using his ring, and forcing them to explain themselves to him.
A tree ring history whose beginning and end dates are not known is called a floating chronology.
It can be anchored by cross-matching a section against another chronology ( tree ring history ) whose dates are known.
See, for example, the case of Meyer v. Mitnick, 625 N. W. 2d 136 ( Michigan, 2001 ), whose ruling found the following reasoning persuasive: " the so-called ' modern trend ' holds that because an engagement ring is an inherently conditional gift, once the engagement has been broken, the ring should be returned to the donor.
Additional theories suggest that either Verica, a British client king of the Roman Empire in the years preceding the Claudian invasion was owner of the palace, or even one Tiberius Claudius Catuarus, whose gold signet ring was recently discovered.
The ring of Gaussian integers is the integral closure of Z in the field of Gaussian rationals Q ( i ) consisting of the complex numbers whose real and imaginary part are both rational.
Picking the right category E is something of a knack: for example, take the given rng R, and make a category E whose objects are rng homomorphisms R → S, with S a ring having a multiplicative identity.
If R is a given commutative ring, then the set of all polynomials in the variable X whose coefficients are in R forms the polynomial ring, denoted R. The same holds true for several variables.
ring and ideals
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.
Formation of the polynomial ring, together with forming factor rings by factoring out ideals, are important tools for constructing new rings out of known ones.
The prime ideals of the ring of integers are the ideals ( 0 ), ( 2 ), ( 3 ), ( 5 ), ( 7 ), ( 11 ), … The fundamental theorem of arithmetic generalizes to the Lasker – Noether theorem, which expresses every ideal in a Noetherian commutative ring as an intersection of primary ideals, which are the appropriate generalizations of prime powers.
Prime ideals are the points of algebro-geometric objects, via the notion of the spectrum of a ring.
For example, prime ideals in the ring of integers of quadratic number fields can be used in proving quadratic reciprocity, a statement that concerns the solvability of quadratic equations
* In any ring R, a maximal ideal is an ideal M that is maximal in the set of all proper ideals of R, i. e. M is contained in exactly 2 ideals of R, namely M itself and the entire ring R. Every maximal ideal is in fact prime.
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.
However, in other rings, the ideals may be distinct from the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements of the ring.
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