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Page "Euclidean domain" ¶ 21
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Z and ring
The unique ring homomorphism from Z to A is determined by the fact that it must send 1 to the identity in A.
* Any ring of characteristic n is a ( Z / nZ )- algebra in the same way.
* Any ring A is an algebra over its center Z ( A ), or over any subring of its center.
In other words, b is a unit in the ring Z / aZ of integers modulo a.
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.
* Z, the ring of integers.
* Z ( where ω is a cube root of 1 ), the ring of Eisenstein integers.
An example of such a finite field is the ring Z / pZ, which is essentially the set of integers from 0 to p − 1 with integer addition and multiplication modulo p. It is also sometimes denoted Z < sub > p </ sub >, but within some areas of mathematics, particularly number theory, this may cause confusion because the same notation Z < sub > p </ sub > is used for the ring of p-adic integers.
Here ( Z / 2Z ) is the polynomial ring of Z / 2Z and ( Z / 2Z )/( T < sup > 2 </ sup >+ T + 1 ) are the equivalence classes of these polynomials modulo T < sup > 2 </ sup >+ T + 1.
However, the same is not true for epimorphisms ; for instance, the inclusion of Z as a ( unitary ) subring of Q is not surjective, but an epimorphic ring homomorphism.
* The ring of p-adic integers is the inverse limit of the rings Z / p < sup > n </ sup > Z ( see modular arithmetic ) with the index set being the natural numbers with the usual order, and the morphisms being " take remainder ".
In the ring Z of integers, the set of prime elements equals the set of irreducible elements, which is
Not every prime ( in Z ) is a Gaussian prime: in the bigger ring Z, 2 factors into the product of the two Gaussian primes ( 1 + i ) and ( 1 − i ).
* In the ring Z of all polynomials with integer coefficients, the ideal generated by 2 and X is a prime ideal.

Z and Gaussian
An example of such a domain is the Gaussian integers Z, that is, the set of complex numbers of the form a + bi where i denotes the imaginary unit and a and b are arbitrary integers.
Rational primes ( i. e. prime elements in Z ) of the form 4k + 3 are Gaussian primes, whereas rational primes of the form 4k + 1 are not.
* Z: the ring of Gaussian integers
The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z.
The prime elements of Z are also known as Gaussian primes.
One should not refer to only these numbers as " the Gaussian primes ", which term refers to all the Gaussian primes, many of which do not lie in Z.
By definition of prime element, if is a Gaussian prime, then it divides ( in Z ) some.
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.
units to some multiple of z, where z is any Gaussian integer ; this turns Z into a Euclidean domain, where
The prime numbers in Z are generalized to irreducible elements in O, and though the unique factorization of elements of O into irreducible elements may hold in some cases ( such as for the Gaussian integers Z ), it may also fail, as in the case of Z
The Gaussian integers Z form the ring of integers of Q ( i ).
The Gauss sum can thus be written as a linear combination of Gaussian periods ( with coefficients χ ( a )); the converse is also true, as a consequence of the orthogonality relations for the group ( Z / nZ )< sup >×</ sup >.
where Z is the Gaussian integer ring, and θ is any non-zero complex number.
For example in the field extension A = Q ( i ) of Gaussian rationals over Q, the integral closure of Z is the ring of Gaussian integers Z and so this is the unique maximal Z-order: all other orders in A are contained in it: for example, we can take the subring of the

Z and integers
* In elementary arithmetic, the set of integers, Z, considered as a group under addition, has a unique nontrivial automorphism: negation.
An example is the " divides " relation between the set of prime numbers P and the set of integers Z, in which every prime p is associated with every integer z that is a multiple of p ( and not with any integer that is not a multiple of p ).
Division of whole numbers can be thought of as a function ; if Z is the set of integers, N < sup >+</ sup > is the set of natural numbers ( except for zero ), and Q is the set of rational numbers, then division is a binary function from Z and N < sup >+</ sup > to Q.
< li > Consider the group ( Z < sub > 6 </ sub >, +), the integers from 0 to 5 with addition modulo 6.
where Z < sub > p </ sub > denotes the p-adic integers.
One of the most familiar groups is the set of integers Z which consists of the numbers
* The group of p-adic integers Z < sub > p </ sup > under addition is profinite ( in fact procyclic ).
* Z: the ring of integers,

Z and .
and enzymes which terminate the action of peptides such as bradykinin and perhaps Substance Z were studied.
If the Z gyro is drifting, a current generated by the autocollimator is delivered to the gyro torquer to cancel the drift.
When the platform is level, **ye is a rotation about the Z axis of the platform Af.
Algeria had also won an Oscar for the movie Z, a political thriller directed by Costa Gavras.
It is conventionally represented by the symbol Z.
The atomic number, Z, should not be confused with the mass number, A, which is the number of nucleons, the total number of protons and neutrons in the nucleus of an atom.
The number of neutrons, N, is known as the neutron number of the atom ; thus, A = Z + N. Since protons and neutrons have approximately the same mass ( and the mass of the electrons is negligible for many purposes ), and the mass defect is usually very small compared to the mass, the atomic mass of an atom is roughly equal to A.
Atoms having the same atomic number Z but different neutron number N, and hence different atomic mass, are known as isotopes.
The conventional symbol Z comes from the German word meaning number / numeral / figure, which prior to the modern synthesis of ideas from chemistry and physics, merely denoted an element's numerical place in the periodic table.
Only after 1915, with the suggestion and evidence that this Z number was also the nuclear charge and a physical characteristic of atoms, did the word and its English equivalent atomic number come into common use.
This placement is consistent with the modern practice of ordering the elements by proton number, Z, but this number was not known or suspected at the time.
Moseley, after discussions with Bohr who was at the same lab ( and who had used Van den Broek's hypothesis in his Bohr model of the atom ), decided to test Van den Broek and Bohr's hypothesis directly, by seeing if spectral lines emitted from excited atoms fit the Bohr theory's demand that the frequency of the spectral lines be proportional to a measure of the square of Z.
To do this, Moseley measured the wavelengths of the innermost photon transitions ( K and L lines ) produced by the elements from aluminum ( Z = 13 ) to gold ( Z = 79 ) used as a series of movable anodic targets inside an x-ray tube.
This led to the conclusion ( Moseley's law ) that the atomic number does closely correspond ( with an offset of one unit for K-lines, in Moseley's work ) to the calculated electric charge of the nucleus, i. e. the proton number Z.
Puttenham, in the time of Elizabeth I of England, wished to start from Elissabet Anglorum Regina ( Elizabeth Queen of the English ), to obtain Multa regnabis ense gloria ( By thy sword shalt thou reign in great renown ); he explains carefully that H is " a note of aspiration only and no letter ", and that Z in Greek or Hebrew is a mere SS.
William Drummond of Hawthornden, in an essay On the Character of a Perfect Anagram, tried to lay down permissible rules ( such as S standing for Z ), and possible letter omissions.
* Stiemerling D. Analysis of a spider and monster phobia, Z. Psychosom Med Psychoanal.
Dial + 64 ( 8 ) 320-1231, from USA / Canada and rest of NANP dial 011-64-8-320-1231 to hear your 3 digit area code & 7 digit local number read back to you from N. Z.

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