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.

* Z, the ring of Gaussian integers.

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.

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 ).

* Z: the ring of Gaussian integers

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.

The Gaussian integers Z form the ring of integers of Q ( i ).

where Z is the Gaussian integer ring, and θ is any non-zero complex number.

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

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

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.

The sum, difference and product of algebraic integers are again algebraic integers, which means that the algebraic integers form a ring.

If K is a number field, its ring of integers is the subring of algebraic integers in K, and is frequently denoted as O < sub > K </ sub >.

As noted in the introduction, Bézout's identity works not only in the ring of integers, but also in any other principal ideal domain ( PID ).

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.

In mathematics, more specifically in abstract algebra and ring theory, a Euclidean domain ( also called a Euclidean ring ) is a ring that can be endowed with a certain structure – namely a Euclidean function, to be described in detail below – which allows a suitable generalization of the Euclidean division of the integers.

This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the ring of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements.

An arbitrary PID has much the same " structural properties " of a Euclidean domain ( or, indeed, even of the ring of integers ), but knowing an explicit algorithm for Euclidean division, and thus also for greatest common divisor computation, gives a concreteness which is useful for algorithmic applications.

Especially, the fact that the integers and any polynomial ring in one variable over a field are Euclidean domains such that the Euclidean division is easily computable is of basic importance in computer algebra.

* Z, the ring of integers.

* Z ( where ω is a cube root of 1 ), the ring of Eisenstein integers.

The most important difference is that fields allow for division ( though not division by zero ), while a ring need not possess multiplicative inverses ; for example the integers form a ring, but 2x = 1 has no solution in integers.

This is the ring of Eisenstein integers, and he proved it has the six units and that it has unique factorization.

Our new large-package ring twister for glass fiber yarns is performing well in our customers' mills.

If you walk into the ring because it is fun to show your dog, he will feel it and give you a good performance!!

`` The reason you are in the ring today is to show your ability to present to any judge the most attractive picture of your dog that the skillful use of your aids can produce.

This is the tale of one John Enright, an American who has accidentally killed a man in the prize ring and is now trying to forget about it in a quiet place where he may become a quiet man.

`` He has married me with a ring of bright water '', begins the Kathleen Raine poem from which Maxwell takes his title, and it is this mystic bond between the human and natural world that the author conveys.

Many theorems which are provable using choice are of an elegant general character: every ideal in a ring is contained in a maximal ideal, every vector space has a basis, and every product of compact spaces is compact.

The field F is algebraically closed if and only if the only irreducible polynomials in the polynomial ring F are those of degree one.

The ceremony of such a blessing is similar in some aspects to the consecration of a bishop, with the new abbot being presented with the mitre, the ring, and the crosier as symbols of office and receiving the laying on of hands and blessing from the celebrant.

The sum, difference, product and quotient of two algebraic numbers is again algebraic ( this fact can be demonstrated using the resultant ), and the algebraic numbers therefore form a field, sometimes denoted by A ( which may also denote the adele ring ) or < span style =" text-decoration: overline ;"> Q </ span >.

In the context of abstract algebra, for example, a mathematical object is an algebraic structure such as a group, ring, or vector space.

Generally speaking, negation is an automorphism of any abelian group, but not of a ring or field.

* A field automorphism is a bijective ring homomorphism from a field to itself.

The same definition holds in any unital ring or algebra where a is any invertible element.

The configuration of six carbon atoms in aromatic compounds is known as a benzene ring, after the simplest possible such hydrocarbon, benzene.

The structure is also illustrated as a circle around the inside of the ring to show six electrons floating around in delocalized molecular orbitals the size of the ring itself.

In aromatic substitution one substituent on the arene ring, usually hydrogen, is replaced by another substituent.

When there is more than one substituent present on the ring, their spatial relationship becomes important for which the arene substitution patterns ortho, meta, and para are devised.

This is seen in, for example, phenol ( C < sub > 6 </ sub > H < sub > 5 </ sub >- OH ), which is acidic at the hydroxyl ( OH ), since a charge on this oxygen ( alkoxide-O < sup >–</ sup >) is partially delocalized into the benzene ring.