Associative rings and algebras Definition
Unit of a ring
Builds on Ring
Definition
Let be a ring. An element is a unit of if there is an element such that . The element is unique. It is the inverse of , with the notation .
Remarks
The units of are and . In a field, every nonzero element is a unit. A real matrix is a unit of the matrix ring if and only if its determinant is not zero. The units of form the group of units of .
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- Group concept
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Where this leads
271 pages build on this one.
- Transcendental element
- Irreducible polynomial
- Uniqueness of finite fields of a given order
- Existence of splitting fields
- Uniqueness of the algebraic closure
- Euclidean algorithm in a Euclidean domain
- Eisenstein's criterion
- Intermediate field
- Splitting field
- Algebraically closed field
- Galois field
- Irreducible element
- Associate elements
- PID implies UFD
- The rationals as the field of fractions of the integers
- Group of units of a ring
- Maximal ideals are prime
- Uniqueness of splitting fields
- The minimal polynomial divides every polynomial vanishing at the element
- Units of the Gaussian integers
- Unique factorisation domain
- Subfield
- Product of primitive polynomials is primitive
- Finite field
- Content of a polynomial
- Subfields of a finite field
- In a PID a nonzero element is irreducible iff it generates a maximal ideal
- Algebraic extension
- Ring homomorphisms from a field are injective
- Tower law
- K[x]/(f) is a field iff f is irreducible
- Order of the Galois group of a Galois extension equals its degree
- Z[√-5] is not a UFD
- Normal extension
- A finite field of order q is the splitting field of x^q - x
- Irreducible elements need not be prime
- Finite subgroups of the multiplicative group of a field are cyclic
- Impossibility of doubling the cube
- Polynomial ring over a UFD is a UFD
- Finite extensions are algebraic
- Galois extension
- A finite integral domain is a field
- The multiplicative group of a finite field is cyclic
- Existence of an algebraic closure
- Impossibility of trisecting the angle
- Prime elements are irreducible in an integral domain
- Greatest common divisor in an integral domain
- Transcendence of pi
- The quaternions
- Minimal polynomial of an algebraic element
- Degree of a field extension
- The order of a finite field is a prime power
- Galois group of a polynomial
- Fixed field
- Reduction modulo p test for irreducibility
- The only ideals of a field are zero and the whole field
- Adjoining a root of a polynomial
- Field generated by adjoining elements
- Algebraic closure
- Prime element
- Separable extension
- Degree of a simple extension is the degree of the minimal polynomial
- The minimal polynomial is irreducible
- Algebraic element
- Finite extension
- Z/pZ is a field for p prime
- Simple extension
- The constructible numbers form a field closed under square roots
- Field
- The algebraic numbers form a field
- Gauss's lemma
- The Galois group has order at most the degree
- In a UFD irreducible elements are prime
- Every element of a finite field of order q satisfies x^q = x
- Wedderburn's little theorem
- Irreducibility of the cyclotomic polynomials over the rationals
- Primitive polynomial
- The degree of a constructible number is a power of two
- Primitive element theorem
- Fundamental theorem of Galois theory
- Existence of finite fields of every prime power order
- An algebraic extension of an algebraic extension is algebraic
- Field of fractions
- Every polynomial has a root in some extension field
- Polynomials of degree two or three are irreducible iff they have no roots
- Frobenius automorphism of a finite field
- Prime subfield
- Bézout's identity in a Euclidean domain
- R/M is a field iff M is maximal
- Field extension
- Division ring
- Impossibility of squaring the circle
- Galois group
- Gauss–Wantzel theorem
- Primitive element of a finite field
- K(a) is isomorphic to K[x] modulo the minimal polynomial
- Unit-regular ring
- Abel–Ruffini theorem
- Regular local rings are UFDs
- Octonions
- Grothendieck's Galois theory
- Formally real field
- Primitive polynomial over a finite field
- Cohn's irreducibility criterion
- List of simple Lie groups
- Field of formal Laurent series
- Wigner D-matrix
- Gelfand–Mazur theorem
- Quaternionic analysis
- Galois group of a prime degree polynomial with two nonreal roots
- Jordan algebra
- Fredholm alternative
- Linear difference equation
- Artin–Schreier theorem
- Perron's irreducibility criterion
- Algebraic independence
- Compact symplectic group
- Characteristic of a field
- Normed field
- Finite Desarguesian planes are Pappian
- 6-j symbol
- Table of Lie groups
- Field trace
- Model theory of algebraically closed fields
- Purely transcendental extension
- Purely inseparable extension
- Radical extension
- Jacobson radical
- Weil conjectures
- Resultant
- Field norm
- Krull domain
- Kaplansky's theorem on PI rings
- Non-Archimedean absolute value
- Poincaré group
- Finite field arithmetic
- Polynomial greatest common divisor
- Representation theory of SU(2)
- Artinian reduced rings are semisimple
- Moduli space
- 3-j symbol
- Lüroth's theorem
- Hua's identity
- Néron–Severi group
- Artin–Schreier theory
- Jacobson's conjecture
- Kaplansky's theorem on projective modules
- Cyclotomic field
- Galois theory
- Cayley–Bacharach theorem
- Tropical variety
- Steinitz's theorem on algebraically closed fields
- Reduction to diagonal form by transvections
- Transcendence basis
- Newton polygon
- Puiseux series
- Quasi-algebraically closed field
- Standard conjectures on algebraic cycles
- Boundary integral method for the Dirichlet problem
- Absolute Galois group
- Function field
- Trace form
- Wigner's classification
- SU(2)
- Factorization of polynomials
- Jacobson density theorem
- Polynomial interpolation
- Separably closed field
- Brauer group
- Noncommutative Gröbner basis
- Lagrange resolvent
- Homotopy continuation for polynomial systems
- Complex p-adic numbers
- Krull–Akizuki theorem
- Galois connection between subgroups and subfields
- Krasner's lemma
- Posner's theorem
- Pythagorean field
- Separable closure
- Nagata's criterion for factoriality
- Normal closure
- Topological field
- Jacobson radical of an Artinian ring is nilpotent
- Capelli's theorem
- Milnor K-theory
- Berlekamp's algorithm
- Transcendental extension
- SL(2,C)
- Jump relations for layer potentials
- Mori–Nagata theorem
- Field homomorphism
- Clean ring
- Spin group
- Conway polynomial
- Field of rational functions
- Berkovich space dynamics
- Cartan–Brauer–Hua theorem
- Real closure
- Bézout's theorem
- Kleinian group
- Casus irreducibilis
- Exceptional Lie group E8
- Kovacic's algorithm
- Galois group of a finite field extension
- Perfect field
- Hilbert's irreducibility theorem
- Dedekind's lemma on independence of characters
- Veblen–Young theorem
- Infinite Galois theory
- GCD domain
- Transcendence degree
- Matrix congruence
- Characterizations of Galois extensions
- Cyclic extension
- Normal basis theorem
- Hilbertian field
- Separably generated extension
- Generic smoothness
- Frobenius theorem on real division algebras
- Lagrange interpolation
- Levitzki's theorem
- Hurwitz quaternions
- Lenagan's theorem
- Categorical theory
- Lorentz group
- Solvability of x^n - a
- Representation theory of the Lorentz group
- Liouvillian function
- Mori domain
- Atomic domain
- Differential Galois theory
- Quintic equation
- Octonionic projective plane
- Tensor product of fields
- Shafarevich's theorem on solvable Galois groups
- 24-cell
- Abelian extension
- Kummer–Artin–Schreier–Witt theory
- Valuation
- Linearly disjoint extensions
- Differential field
- Discriminant of a field extension
- Hahn series
- Cyclotomic extension
- Étale fundamental group
- Pin group
- Complex structure on the six-sphere
- Krull topology
- Perfect closure
- Tsen's theorem
- Galois' theorem on solvability
- Nyström method
- Amitsur's theorem on the radical of polynomial rings
- Hopkins–Levitzki theorem
- Existence of transcendence bases
- Prüfer domain
- Impossibility of the classical construction problems
- Compositum of fields
- Artin's theorem on fixed fields
- Fredholm's theorem
- Separable degree
- Factorization of polynomials over finite fields
- Divisor class group
- Artin–Schreier theorem on formally real fields
- Hilbert's Theorem 90
- Inverse Galois problem
- Valuation ring
- Exceptional Lie group G2
- Solvable by radicals
- Newton–Puiseux theorem
- Discrete valuation
Further reading
- Web https://en.wikipedia.org/wiki/Unit_(ring_theory), lead section
About this page
| Tag | 019NFGFPR5 |
|---|---|
| URI | m2://019NFGFPR5 |
| Kind | definition |
| Writing | W1 stated |
| MSC | 16U60 |
| Defines | Unit of a ring |
| Uses | Ring |
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What this page needs for W2
- edge instance_of in: one example (0 of 1)
- edge not_instance_of in: one non-example that fails one clause (0 of 1)
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