# Ring (mathematics)

**ringringsassociative ringassociative ringsring with unityunitalunital ringring axiomsring theory(unital) ring**

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.wikipedia

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### Abstract algebra

**algebraalgebraicmodern algebra**

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.

Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras.

### Algebraic structure

**algebraic structuresunderlying setalgebraic system**

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.

Examples of algebraic structures include groups, rings, fields, and lattices.

### Ring theory

**ringringsring-theoretic**

As a result, commutative ring theory, commonly known as commutative algebra, is a key topic in ring theory.

In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers.

### Series (mathematics)

**infinite seriesseriespartial sum**

Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series, matrices and functions.

Generally, the terms of a series come from a ring, often the field \mathbb R of the real numbers or the field \mathbb C of the complex numbers.

### Algebraic number theory

**placefinite placealgebraic**

Its development has been greatly influenced by problems and ideas occurring naturally in algebraic number theory and algebraic geometry.

These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.

### Group ring

**group algebraintegral group ringgroup C*-algebra**

Examples of noncommutative rings include the ring of n × n real square matrices with n ≥ 2, group rings in representation theory, operator algebras in functional analysis, rings of differential operators in the theory of differential operators, and the cohomology ring of a topological space in topology.

In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group.

### Abelian group

**abelianabelian groupsadditive group**

A ring is an abelian group with a second binary operation that is associative, is distributive over the abelian group operation, and has an identity element (this last property is not required by some authors, see ).

Generally, the multiplicative notation is the usual notation for groups, while the additive notation is the usual notation for modules and rings.

### Identity element

**identityneutral elementleft identity**

A ring is an abelian group with a second binary operation that is associative, is distributive over the abelian group operation, and has an identity element (this last property is not required by some authors, see ).

This concept is used in algebraic structures such as groups and rings.

### Ring of integers

**rings of integersintegral basisnumber ring**

Examples of commutative rings include the set of integers equipped with the addition and multiplication operations, the set of polynomials equipped with their addition and multiplication, the coordinate ring of an affine algebraic variety, and the ring of integers of a number field.

In mathematics, the ring of integers of an algebraic number field K is the ring of all integral elements contained in K.

### Emmy Noether

**NoetherAmalie "Emmy" NoetherE. Noether**

Key contributors include Dedekind, Hilbert, Fraenkel, and Noether.

As one of the leading mathematicians of her time, she developed the theories of rings, fields, and algebras.

### Polynomial ring

**polynomial algebraring of polynomialspolynomial**

Rings were first formalized as a generalization of Dedekind domains that occur in number theory, and of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory.

In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field.

### Integer

**integersintegralZ**

By extension from the integers, the abelian group operation is called addition and the second binary operation is called multiplication. Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called commutative rings. The most familiar example of a ring is the set of all integers, \mathbb{Z}, consisting of the numbers

The integers form the smallest group and the smallest ring containing the natural numbers.

### Commutative ring

**commutativecommutative rings rings**

Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called commutative rings.

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative.

### Number

**number systemnumericalnumbers**

The most familiar example of a ring is the set of all integers, \mathbb{Z}, consisting of the numbers

Today, number systems are considered important special examples of much more general categories such as rings and fields, and the application of the term "number" is a matter of convention, without fundamental significance.

### Zero ring

**trivial ringnonzerotrivial**

A (non-trivial) commutative ring such that every nonzero element has a multiplicative inverse is called a field.

In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element.

### Matrix ring

**matrix algebramatrix algebrasring of matrices**

More generally, for any ring R, commutative or not, and any nonnegative integer n, one may form the ring of n-by-n matrices with entries in R: see Matrix ring.

In abstract algebra, a matrix ring is any collection of matrices over some ring R that form a ring under matrix addition and matrix multiplication.

### Matrix multiplication

**matrix productmultiplicationproduct**

With the operations of matrix addition and matrix multiplication, this set satisfies the above ring axioms.

In mathematics, matrix multiplication or matrix product is a binary operation that produces a matrix from two matrices with entries in a field, or, more generally, in a ring or even a semiring.

### Rng (algebra)

**rngnon-unital ringrng of square zero**

Authors who follow this convention sometimes refer to a structure satisfying all the axioms except the requirement that there exists a multiplicative identity element as a rng (commonly pronounced rung) and sometimes as a pseudo-ring.

In mathematics, and more specifically in abstract algebra, a rng (or pseudo-ring or non-unital ring) is an algebraic structure satisfying the same properties as a ring, without assuming the existence of a multiplicative identity.

### Cohomology ring

**cup-length**

Examples of noncommutative rings include the ring of n × n real square matrices with n ≥ 2, group rings in representation theory, operator algebras in functional analysis, rings of differential operators in the theory of differential operators, and the cohomology ring of a topological space in topology.

In mathematics, specifically algebraic topology, the cohomology ring of a topological space X is a ring formed from the cohomology groups of X together with the cup product serving as the ring multiplication.

### Boolean ring

**Boolean ringsring**

In mathematics, a Boolean ring R is a ring for which x 2 = x for all x in R, such as the ring of integers modulo 2.

### Module (mathematics)

**modulemodulessubmodule**

The monoid action of a ring R on an abelian group is simply an R-module.

A module over a ring is a generalization of the notion of vector space over a field, wherein the corresponding scalars are the elements of an arbitrary given ring (with identity) and a multiplication (on the left and/or on the right) is defined between elements of the ring and elements of the module.

### Operator algebra

**operator algebrasalgebraalgebra of bounded singular integral operators**

Examples of noncommutative rings include the ring of n × n real square matrices with n ≥ 2, group rings in representation theory, operator algebras in functional analysis, rings of differential operators in the theory of differential operators, and the cohomology ring of a topological space in topology.

In general operator algebras are non-commutative rings.

### Nilpotent

**nilpotent elementnilpotencenilpotent endomorphism**

A nilpotent element is an element a such that a^n = 0 for some n > 0.

In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that x n = 0.

### Matrix (mathematics)

**matrixmatricesmatrix theory**

Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series, matrices and functions.

Abstract algebra uses matrices with entries in more general fields or even rings, while linear algebra codifies properties of matrices in the notion of linear maps.

### Free module

**freefree vector spacefree ''R''-module**

Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules.