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 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.

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.

In the brief moment I had to talk to them before I took my post on the ring of defenses, I indicated I was sickened by the methods men employed to live and trade on the river.

For a blood-chilling ring of terror to the very sound of his name was the tool he needed for the job he'd promised to do.

Opposite every gate was a hitching post or a stone carriage-step, set with a rusty iron ring for tying a horse.

When they were first written, there was evidently no thought of their being published, and those which refer to the writer's love for Mrs. Meynell particularly have the ring of truth.

Both abolition of war and new techniques of production, particularly robot factories, greatly increase the world's wealth, a situation described in the following passage, which has the true utopian ring: `` Everything was so cheap that the necessities of life were free, provided as a public service by the community, as roads, water, street lighting and drainage had once been.

Her mother, now dead, was my good friend and when she came to tell us about her plans and to show off her ring I had a sobering wish to say something meaningful to her, something her mother would wish said.

As the Juniors entered the ring, Mr. Spring, the announcer, stated over the public-address system that this was the 28th year that Westminster has held the Finals of the Junior Competition.

After passing Af through DEAE-cellulose, the titer of antibodies to WTV in the specific fraction was 1: 4 of the titer before such passage ( precipitin ring tests by R. F. Whitcomb ) ; ;

( Jelke later served 21 months when he was found guilty of masterminding a ring of high-priced call girls.

Just how many sub secrets were being handed over when the ring, watched for six months, was broken remained untold.

The spy ring also was particularly interested in ASDIC, the underwater equipment for detecting submarines, it was testified.

Six radiomen told how, twice on two days after the ring was nabbed, a transmitter near Moscow was heard calling, using signals, times and wavelengths specified on codes found hidden in cigaret lighters in Lonsdale's apartment and the Krogers' house and also fastened to the transmitter lid.

instead of a necktie he wore a leather bolo drawn through a golden ring in which was set a lump of pale pure jade.

There was no ring of animal corpses.

Sakharov then tested a MK-driven " plasma cannon " where a small aluminium ring was vaporized by huge eddy currents into a stable, self-confined toroidal plasmoid and was accelerated to 100 km / s.

Australian Rules footballer Daniel Chick elected to have his left ring finger amputated as chronic pain and injury was limiting his performance.

The shaft was fitted into the socket of the fore shaft and a bone ring was then placed over the joint to hold the two pieces together, as well as, protecting the wooden shaft from splitting.

The Throwing lance usually consisted of three parts: a wooden shaft, a bone ring or belt, and the compound head that was made with a barbed bonehead and a stone tip.

The bone ring was designed to break after impact so that the shaft could be used again for another kill.

' Within the beautiful red rose, there was a ring.

Descartes recognized that there would be a real difference, however, between a situation in which a body with movable parts and originally at rest with respect to a surrounding ring was itself accelerated to a certain angular velocity with respect to the ring, and another situation in which the surrounding ring was given a contrary acceleration with respect to the central object.