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, the ring of Gaussian 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.

NB: Z < sup > Y </ sup > is the set of all functions from Y to Z

For example, the division example above may also be interpreted as a partial binary function from Z and N to Q, where N is the set of all natural numbers, including zero.

A binary operation is a binary function where the sets X, Y, and Z are all equal ; binary operations are often used to define algebraic structures.

In linear algebra, a bilinear transformation is a binary function where the sets X, Y, and Z are all vector spaces and the derived functions f < sup > x </ sup > and f < sub > y </ sub > are all linear transformations.

For example, intervals, where takes all integer values in Z, cover R but there is no finite subcover.

But there could be a third variable ( Z ) that influences ( Y ), and X might not be the true cause at all.

Gravitons are postulated because of the great success of quantum field theory ( in particular, the Standard Model ) at modeling the behavior of all other known forces of nature as being mediated by elementary particles: electromagnetism by the photon, the strong interaction by the gluons, and the weak interaction by the W and Z bosons.

There is one more automorphism with this property: multiplying all elements of Z < sub > 7 </ sub > by 5, modulo 7.

By placing the molecules in wells in the gel and applying an electric field, the molecules will move through the matrix at different rates, determined largely by their mass when the charge to mass ratio ( Z ) of all species is uniform, toward the ( negatively charged ) cathode if positively charged or toward the ( positively charged ) anode if negatively charged.

and Z. evolved into the Cukor-Kondolf Stock Company, a troupe that included Louis Calhern, Ilka Chase, Phyllis Povah, Frank Morgan, Reginald Owen, Elizabeth Patterson and Douglass Montgomery, all of whom would work with Cukor in later years in Hollywood.

With regard to the second and / or third letters in the prefixes in the list below, if the country in question is allocated all callsigns with A to Z in that position, then that country can also use call signs with the digits 0 to 9 in that position.

Other computers, all numbered with a leading Z, up to Z43, were built by Zuse and his company.

* See List of musicals: A to L and List of musicals: M to Z for a list of musicals in alphabetical order ; note that not all of these have been made into films.

It is the inverse limit of the finite groups Z / p < sup > n </ sup > Z where n ranges over all natural numbers and the natural maps Z / p < sup > n </ sup > Z → Z / p < sup > m </ sup > Z ( n ≥ m ) are used for the limit process.

* Reduced form, in statistics, an equation which relates the endogenous variable X to all the available exogenous variables, both those included in the regression of interest ( W ) and the instruments ( Z )

C < sub > 1 </ sub > is the trivial group containing only the identity operation, which occurs when the figure has no symmetry at all, for example the letter F. C < sub > 2 </ sub > is the symmetry group of the letter Z, C < sub > 3 </ sub > that of a triskelion, C < sub > 4 </ sub > of a swastika, and C < sub > 5 </ sub >, C < sub > 6 </ sub > etc.

Roughly T < sup > 2 </ sup >+ T + 1 = 0 so that T < sup > 2 </ sup >= T + 1 ( since − 1 = 1 in Z / 2Z ) and hence the elements of ( Z / 2Z )/( T < sup > 2 </ sup >+ T + 1 ) are the polynomials of degree up to 1 with coefficients in Z / 2Z, i. e. the set

So for k even the polynomials Z ( E, T ) have no zeros in this region, or in other words the eigenvalues of Frobenius on the stalks of E < sup > k </ sup > have absolute value at most q < sup > k ( d − 1 )/ 2 + 1 </ sup >.

For each set S of polynomials in k ..., x < sub > n </ sub >, define the zero-locus Z ( S ) to be the set of points in A < sup > n </ sup > on which the functions in S simultaneously vanish, that is to say

A different example is that of the Laurent polynomials over a ring R: these are nothing more or less than the group ring of the infinite cyclic group Z over R.

A ring of symmetric polynomials can be defined over any commutative ring R, and will be denoted Λ < sub > R </ sub >; the basic case is for R = Z.

Specifically, assume we have a nonsingular variety X, and n hypersurfaces Z < sub > 1 </ sub >, ..., Z < sub > n </ sub > which have local equations f < sub > 1 </ sub >, ..., f < sub > n </ sub > near x for polynomials f < sub > i </ sub >( t < sub > 1 </ sub >, ..., t < sub > n </ sub >), such that the following hold:

These sequences may be represented as coefficients of irreducible polynomials in a polynomial ring over Z / 2Z.

* Primitivity statement: The set of primitive polynomials in Z is closed under multiplication: if P and Q are primitive polynomials then so is their product PQ.

An elementary proof of the statement that the product of primitive polynomials over Z is again primitive can be given as follows.

We formulate this proof directly for the case of polynomials over a UFD R, which is hardly different from its special case for R = Z.