Logo
Unionpedia
Communication
Get it on Google Play
New! Download Unionpedia on your Android™ device!
Free
Faster access than browser!
 

Unit (ring theory)

Index Unit (ring theory)

In mathematics, an invertible element or a unit in a (unital) ring is any element that has an inverse element in the multiplicative monoid of, i.e. an element such that The set of units of any ring is closed under multiplication (the product of two units is again a unit), and forms a group for this operation. [1]

51 relations: Additive identity, Additive inverse, Adjoint functors, Algebra over a field, Algebraic number field, Cardinality, Category of groups, Category of rings, Complement (set theory), Coprime integers, Dirichlet's unit theorem, Division ring, Domain (ring theory), Equivalence relation, Field (mathematics), Finitely generated module, Formal power series, Functor, General linear group, Graduate Texts in Mathematics, Group (mathematics), Group action, Group homomorphism, Group ring, Identity matrix, If and only if, Integer, Integral domain, Inverse element, Invertible matrix, John Wiley & Sons, Local ring, Mathematics, Monoid, Multiplication, Multiplicative group of integers modulo n, Nilpotent, Polynomial ring, Quadratic field, Real number, Ring (mathematics), Ring homomorphism, Ring of integers, Root of unity, S-unit, Springer Science+Business Media, Square matrix, X, Zero divisor, Zero ring, ..., 1. Expand index (1 more) »

Additive identity

In mathematics the additive identity of a set which is equipped with the operation of addition is an element which, when added to any element x in the set, yields x. One of the most familiar additive identities is the number 0 from elementary mathematics, but additive identities occur in other mathematical structures where addition is defined, such as in groups and rings.

New!!: Unit (ring theory) and Additive identity · See more »

Additive inverse

In mathematics, the additive inverse of a number is the number that, when added to, yields zero.

New!!: Unit (ring theory) and Additive inverse · See more »

Adjoint functors

In mathematics, specifically category theory, adjunction is a possible relationship between two functors.

New!!: Unit (ring theory) and Adjoint functors · See more »

Algebra over a field

In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product.

New!!: Unit (ring theory) and Algebra over a field · See more »

Algebraic number field

In mathematics, an algebraic number field (or simply number field) F is a finite degree (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.

New!!: Unit (ring theory) and Algebraic number field · See more »

Cardinality

In mathematics, the cardinality of a set is a measure of the "number of elements of the set".

New!!: Unit (ring theory) and Cardinality · See more »

Category of groups

In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms.

New!!: Unit (ring theory) and Category of groups · See more »

Category of rings

In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (preserving the identity).

New!!: Unit (ring theory) and Category of rings · See more »

Complement (set theory)

In set theory, the complement of a set refers to elements not in.

New!!: Unit (ring theory) and Complement (set theory) · See more »

Coprime integers

In number theory, two integers and are said to be relatively prime, mutually prime, or coprime (also written co-prime) if the only positive integer (factor) that divides both of them is 1.

New!!: Unit (ring theory) and Coprime integers · See more »

Dirichlet's unit theorem

In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet.

New!!: Unit (ring theory) and Dirichlet's unit theorem · See more »

Division ring

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

New!!: Unit (ring theory) and Division ring · See more »

Domain (ring theory)

In mathematics, and more specifically in algebra, a domain is a nonzero ring in which implies or.

New!!: Unit (ring theory) and Domain (ring theory) · See more »

Equivalence relation

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.

New!!: Unit (ring theory) and Equivalence relation · See more »

Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

New!!: Unit (ring theory) and Field (mathematics) · See more »

Finitely generated module

In mathematics, a finitely generated module is a module that has a finite generating set.

New!!: Unit (ring theory) and Finitely generated module · See more »

Formal power series

In mathematics, a formal power series is a generalization of a polynomial, where the number of terms is allowed to be infinite; this implies giving up the possibility of replacing the variable in the polynomial with an arbitrary number.

New!!: Unit (ring theory) and Formal power series · See more »

Functor

In mathematics, a functor is a map between categories.

New!!: Unit (ring theory) and Functor · See more »

General linear group

In mathematics, the general linear group of degree n is the set of invertible matrices, together with the operation of ordinary matrix multiplication.

New!!: Unit (ring theory) and General linear group · See more »

Graduate Texts in Mathematics

Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag.

New!!: Unit (ring theory) and Graduate Texts in Mathematics · See more »

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

New!!: Unit (ring theory) and Group (mathematics) · See more »

Group action

In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.

New!!: Unit (ring theory) and Group action · See more »

Group homomorphism

In mathematics, given two groups, (G, ∗) and (H, ·), a group homomorphism from (G, ∗) to (H, ·) is a function h: G → H such that for all u and v in G it holds that where the group operation on the left hand side of the equation is that of G and on the right hand side that of H. From this property, one can deduce that h maps the identity element eG of G to the identity element eH of H, and it also maps inverses to inverses in the sense that Hence one can say that h "is compatible with the group structure".

New!!: Unit (ring theory) and Group homomorphism · See more »

Group ring

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.

New!!: Unit (ring theory) and Group ring · See more »

Identity matrix

In linear algebra, the identity matrix, or sometimes ambiguously called a unit matrix, of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere.

New!!: Unit (ring theory) and Identity matrix · See more »

If and only if

In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.

New!!: Unit (ring theory) and If and only if · See more »

Integer

An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

New!!: Unit (ring theory) and Integer · See more »

Integral domain

In mathematics, and specifically in abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero.

New!!: Unit (ring theory) and Integral domain · See more »

Inverse element

In abstract algebra, the idea of an inverse element generalises concepts of a negation (sign reversal) in relation to addition, and a reciprocal in relation to multiplication.

New!!: Unit (ring theory) and Inverse element · See more »

Invertible matrix

In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular or nondegenerate) if there exists an n-by-n square matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication.

New!!: Unit (ring theory) and Invertible matrix · See more »

John Wiley & Sons

John Wiley & Sons, Inc., also referred to as Wiley, is a global publishing company that specializes in academic publishing.

New!!: Unit (ring theory) and John Wiley & Sons · See more »

Local ring

In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime.

New!!: Unit (ring theory) and Local ring · See more »

Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

New!!: Unit (ring theory) and Mathematics · See more »

Monoid

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element.

New!!: Unit (ring theory) and Monoid · See more »

Multiplication

Multiplication (often denoted by the cross symbol "×", by a point "⋅", by juxtaposition, or, on computers, by an asterisk "∗") is one of the four elementary mathematical operations of arithmetic; with the others being addition, subtraction and division.

New!!: Unit (ring theory) and Multiplication · See more »

Multiplicative group of integers modulo n

In modular arithmetic, the integers coprime (relatively prime) to n from the set \ of n non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n. Equivalently, the elements of this group can be thought of as the congruence classes, also known as residues modulo n, that are coprime to n. Hence another name is the group of primitive residue classes modulo n. In the theory of rings, a branch of abstract algebra, it is described as the group of units of the ring of integers modulo n. Here units refers to elements with a multiplicative inverse, which in this ring are exactly those coprime to n. This group, usually denoted (\mathbb/n\mathbb)^\times, is fundamental in number theory.

New!!: Unit (ring theory) and Multiplicative group of integers modulo n · See more »

Nilpotent

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

New!!: Unit (ring theory) and Nilpotent · See more »

Polynomial ring

In mathematics, especially in the field of abstract 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.

New!!: Unit (ring theory) and Polynomial ring · See more »

Quadratic field

In algebraic number theory, a quadratic field is an algebraic number field K of degree two over Q, the rational numbers.

New!!: Unit (ring theory) and Quadratic field · See more »

Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

New!!: Unit (ring theory) and Real number · See more »

Ring (mathematics)

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

New!!: Unit (ring theory) and Ring (mathematics) · See more »

Ring homomorphism

In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the structure.

New!!: Unit (ring theory) and Ring homomorphism · See more »

Ring of integers

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

New!!: Unit (ring theory) and Ring of integers · See more »

Root of unity

In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power.

New!!: Unit (ring theory) and Root of unity · See more »

S-unit

In mathematics, in the field of algebraic number theory, an S-unit generalises the idea of unit of the ring of integers of the field.

New!!: Unit (ring theory) and S-unit · See more »

Springer Science+Business Media

Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

New!!: Unit (ring theory) and Springer Science+Business Media · See more »

Square matrix

In mathematics, a square matrix is a matrix with the same number of rows and columns.

New!!: Unit (ring theory) and Square matrix · See more »

X

X (named ex, plural exes) is the 24th and antepenultimate letter in the modern English alphabet and the ISO basic Latin alphabet.

New!!: Unit (ring theory) and X · See more »

Zero divisor

In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero such that, or equivalently if the map from to that sends to is not injective.

New!!: Unit (ring theory) and Zero divisor · See more »

Zero ring

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

New!!: Unit (ring theory) and Zero ring · See more »

1

1 (one, also called unit, unity, and (multiplicative) identity) is a number, numeral, and glyph.

New!!: Unit (ring theory) and 1 · See more »

Redirects here:

Group of units, Invertible element, Unit (abstract algebra), Unit (algebra), Unit (mathematics), Unit group.

References

[1] https://en.wikipedia.org/wiki/Unit_(ring_theory)

OutgoingIncoming
Hey! We are on Facebook now! »