* Generating set of an ideal ( ring theory ):

In 1930, England captain Douglas Jardine, together with Nottinghamshire's captain Arthur Carr and his bowlers Harold Larwood and Bill Voce, developed a variant of leg theory in which the bowlers bowled fast, short-pitched balls that would rise into the batsman's body, together with a heavily stacked ring of close fielders on the leg side.

His mathematical specialties were noncommutative ring theory and computational algebra and its applications, including automated theorem proving in geometry.

* Conductor ( ring theory ), an ideal of a ring that measures how far it is from being integrally closed

There are different definitions used in group theory and ring theory.

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.

Another definition of the GCD is helpful in advanced mathematics, particularly ring theory.

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.

Even though the set may be the same, the same function might be a homomorphism, say, in group theory ( sets with a single operation ) but not in ring theory ( sets with two related operations ), because it fails to preserve the additional operation that ring theory considers.

In the language of category theory it is a morphism in the category of modules over a given ring.

In ring theory, the notion of number is generally replaced with that of ideal.

* Radical of a ring, in ring theory, a radical of a ring is an ideal of " bad " elements of the ring

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.

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

In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible.

In abstract algebra, a field is a commutative ring which contains a multiplicative inverse for every nonzero element, equivalently a ring whose nonzero elements form an abelian group under multiplication.

In abstract algebra, an integral domain is a commutative ring that has no zero divisors, and which is not the trivial ring

In abstract algebra, a monoid ring is a new ring constructed from some other ring and a monoid.

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring.

In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec ( R ), is the set of all proper prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space.

In abstract algebra, a congruence relation ( or simply congruence ) is an equivalence relation on an algebraic structure ( such as a group, ring, or vector space ) that is compatible with the structure.

From the point of view of abstract algebra, congruence modulo is a congruence relation on the ring of integers, and arithmetic modulo occurs on the corresponding quotient ring.

In the branch of mathematics known as abstract algebra, a ring is an algebraic concept abstracting and generalizing the algebraic structure of the integers, specifically the two operations of addition and multiplication.

In abstract algebra, two nonzero elements and of a ring are respectively called a left zero divisor and a right zero divisor if ; this is a partial case of divisibility in rings.

It occurs in the proofs of several theorems of crucial importance, for instance the Hahn – Banach theorem in functional analysis, the theorem that every vector space has a basis, Tychonoff's theorem in topology stating that every product of compact spaces is compact, and the theorems in abstract algebra that every nonzero ring has a maximal ideal and that every field has an algebraic closure.

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

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

* 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.

* Any commutative ring R is an algebra over itself, or any subring of R.

This loop of units in an alternative ring or algebra is analogous to the group of units in an associative ring or algebra.

Every Boolean algebra ( A, ∧, ∨) gives rise to a ring ( A, +, ·) by defining a + b := ( a ∧ ¬ b ) ∨ ( b ∧ ¬ a ) = ( a ∨ b ) ∧ ¬( a ∧ b ) ( this operation is called symmetric difference in the case of sets and XOR in the case of logic ) and a · b := a ∧ b. The zero element of this ring coincides with the 0 of the Boolean algebra ; the multiplicative identity element of the ring is the 1 of the Boolean algebra.

Conversely, if a Boolean ring A is given, we can turn it into a Boolean algebra by defining x ∨ y := x + y + ( x · y ) and x ∧ y := x · y.

Since these two constructions are inverses of each other, we can say that every Boolean ring arises from a Boolean algebra, and vice versa.

( The definition of a homomorphism depends on the type of algebraic structure ; see, for example: group homomorphism, ring homomorphism, and linear operator ).

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

Starting with a ring A, we get a unital associative R-algebra by providing a ring homomorphism whose image lies in the center of A.

A homomorphism between two associative R-algebras is an R-linear ring homomorphism.

The unique ring homomorphism from Z to A is determined by the fact that it must send 1 to the identity in A.

For the case of a non-commutative base ring R and a right module M < sub > R </ sub > and a left module < sub > R </ sub > N, we can define a bilinear map, where T is an abelian group, such that for any n in N, is a group homomorphism, and for any m in M, is a group homomorphism too, and which also satisfies

A ring homomorphism of commutative rings determines a morphism of Kähler differentials which sends an element dr to d ( f ( r )), the exterior differential of f ( r ).

* A ring homomorphism is a homomorphism between two rings.

For example, a ring possesses both addition and multiplication, and a homomorphism from the ring to the ring is a function such that

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.

This inclusion thus also is an example of a ring homomorphism which is both mono and epi, but not iso.

If a is an idempotent of the endomorphism ring End < sub > R </ sub >( M ), then the endomorphism is an R module involution of M. That is, f is an R homomorphism such that f < sup > 2 </ sup > is the identity endomorphism of M.

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.