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 ).

Before the Juniors entered the ring the Steward announced that after all Juniors had moved their dogs around the ring and set them up, they could relax with their dogs.

After the judge moved all the dogs individually, she selected several from the group and placed them in the center of the ring.

If a is algebraic over K, then K, the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. In the special case where K = Q is the field of rational numbers, Q is an example of an algebraic number field.

This ring has the property that a · a = a for all a in A ; rings with this property are called Boolean rings.

There were twelve rules in all, and they specified that fights should be " a fair stand-up boxing match " in a 24-foot-square or similar ring.

When there are no enemy discs on the board, many ( but not all ) rules also state that a player must shoot for the centre of the board, and a shot disc must finish either completely inside the 15-point guarded ring line, or ( depending on the specifics of the rules ) be inside or touching this line.

* The spectrum of any commutative ring with the Zariski topology ( that is, the set of all prime ideals ) is compact, but never Hausdorff ( except in trivial cases ).

With the 9 main keys, ( operated by the index, middle, and ring fingers ), 2 prefix keys and one delete key, the EkaPad can produce all the inputs of a standard qwerty keyboard with one, two, and a few three finger chords.

In his acceptance speech, Powell reminded Americans that " It is for America, the Land of the Free and the Home of the Brave, to help freedom ring across the globe, unto all the peoples thereof.

The first inductee was Bob Lilly in 1975 and by 2005, the ring contained 17 names, all former Dallas players except for one head coach and one general manager / president.

For square matrices with entries in a non-commutative ring, for instance the quaternions, there is no unique definition for the determinant, and no definition that has all the usual properties of determinants over commutative rings.

Stated differently, a ring is a division ring if and only if the group of units is the set of all non-zero elements.

When his father's ring was sent to him, he begged that his father would show mercy to his mother, and that all his companions would plead with Henry to set her free.

Some authors also require the domain of the Euclidean function be the entire ring R ; this can always be accommodated by adding 1 to the values at all nonzero elements, and defining the function to be 0 at the zero element of R, but the result is somewhat awkward in the case of K. The definition is sometimes generalized by allowing the Euclidean function to take its values in any well-ordered set ; this weakening does not affect the most important implications of the Euclidean property.

The property ( EF1 ) can be restated as follows: for any principal ideal I of R with nonzero generator b, all nonzero classes of the quotient ring R / I have a representative r with.

As in the simple cases above, one may enumerate all distinct four-colorings of the ring ; any coloring that can be extended without modification to a coloring of the configuration is called initially good.

Surrounding all is a ring or horseshoe-shaped layer of raw coal, usually kept damp and tightly packed to maintain the shape of the fire's heart and to keep the coal from burning directly so that it " cooks " into coke first.

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.

In a ring all of whose ideals are principal ( a principal ideal domain or PID ), this ideal will be identical with the set of multiples of some ring element d ; then this d is a greatest common divisor of a and b. But the ideal ( a, b ) can be useful even when there is no greatest common divisor of a and b. ( Indeed, Ernst Kummer used this ideal as a replacement for a gcd in his treatment of Fermat's Last Theorem, although he envisioned it as the set of multiples of some hypothetical, or ideal, ring element d, whence the ring-theoretic term.

< http :// www. chm. bris. ac. uk / motm / hemoglobin / hemoglobjm. htm >.</ ref > The iron ion, which is the site of oxygen binding, coordinates with the four nitrogens in the center of the ring, which all lie in one plane.

However, this all changed in 450 when Honoria, sister of the Western Roman Emperor Valentinian III, sent Attila a ring and requested his help to escape her betrothal to a senator.

In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F, the ring of polynomials in the variable x with coefficients in F.

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

* K, the ring of polynomials over a field K. For each nonzero polynomial P, define f ( P ) to be the degree of P.

In mathematics, specifically commutative algebra, Hilbert's basis theorem states that every ideal in the ring of multivariate polynomials over a Noetherian ring is finitely generated.

Although different as expressions, these two expressions are equal in the ring of the polynomials in the indeterminate x with integer coefficients.

Thus the set of all polynomials with coefficients in the ring R forms itself a ring, the ring of polynomials over R, which is denoted by R. The map from R to R sending r to rX < sup > 0 </ sup > is an injective homomorphism of rings, by which R is viewed as a subring of R. If R is commutative, then R is an algebra over R.

The polynomials q and r are uniquely determined by f and g. This is called Euclidean division, division with remainder or polynomial long division and shows that the ring F is a Euclidean domain.

* If R denotes the ring CY of polynomials in two variables with complex coefficients, then the ideal generated by the polynomial Y < sup > 2 </ sup > − X < sup > 3 </ sup > − X − 1 is a prime ideal ( see elliptic curve ).

* Z: the ring of all polynomials with integer coefficients --- it is not principal because the ideal generated by 2 and X is an example of an ideal that cannot be generated by a single polynomial.

The converse does not hold since for any field K, K is a UFD but is not a PID ( to prove this look at the ideal generated by It is not the whole ring since it contains no polynomials of degree 0, but it cannot be generated by any one single element ).

More generally, if R is any ring, the set of all polynomials with coefficients in R can be equipped with the structure of a ring and is denoted by R.

By induction, Hilbert's basis theorem establishes that, the ring of all polynomials in n variables with coefficients in, is a Noetherian ring.

If R is a ring, let R denote the ring of polynomials in the indeterminate X over R. Hilbert proved that if R is " not too large ", in the sense that if R is Noetherian, the same must be true for R. Formally,