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.

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.

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

* 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,

Every module over a division ring has a basis ; linear maps between finite-dimensional modules over a division ring can be described by matrices, and the Gaussian elimination algorithm remains applicable.

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.

If R is a Euclidean domain in which euclidean division is given algorithmically ( as is the case for instance when R = F where F is a field, or when R is the ring of Gaussian integers ), then greatest common divisors can be computed using a form of the Euclidean algorithm based on the division procedure.

The ring of Gaussian integers was introduced by Carl Friedrich Gauss in his second monograph on quartic reciprocity ( 1832 ) ( see ).

In 1841 he generalized his arithmetic progressions theorem from integers to the ring of Gaussian integers.

Any such complex torus has the Gaussian integers as endomorphism ring.

Q ( i ), so O < sub > K </ sub > is simply Z, and O < sub > L </ sub > = Z is the ring of Gaussian integers.

In his second monograph on biquadratic reciprocity, Gauss used a fourth-power lemma to derive the formula for the biquadratic character of 1 + i in Z, the ring of Gaussian integers.