Commutative algebra Definition
Commutative ring
Builds on Ring
Definition
A commutative ring is a ring such that for all .
Remarks
The integers , the rational numbers and the polynomials in one variable over are commutative rings. The ring of real matrices is not commutative. Every field is a commutative ring.
Reading path
To understand this page from the ground up, read these 3 pages in order. The foundations come first.
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- Group concept
- Ring definition
- Commutative ring definition
Where this leads
324 pages build on this one.
- The integers are a Euclidean domain
- The characteristic of an integral domain is zero or prime
- Principal ideal
- 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
- Ideal generated by a subset
- Freshman's dream
- Norm on the Gaussian integers
- Divisibility in a commutative ring
- Galois field
- Irreducible element
- Associate elements
- PID implies UFD
- The rationals as the field of fractions of the integers
- Maximal ideals are prime
- Uniqueness of splitting fields
- Symmetric polynomial
- Z[x] is not a principal ideal domain
- Prime ideal
- 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
- Integral domain
- Tower law
- Power sum symmetric polynomials
- R/P is an integral domain iff P is prime
- K[x]/(f) is a field iff f is irreducible
- R[x] is an integral domain iff R is
- Order of the Galois group of a Galois extension equals its degree
- Z[√-5] is not a UFD
- Normal extension
- Discriminant of a polynomial
- A finite field of order q is the splitting field of x^q - x
- Polynomials in several variables
- 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
- K[x] is a Euclidean domain
- 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
- Frobenius endomorphism
- Euclidean domain
- Cancellation in an integral domain
- Minimal polynomial of an algebraic element
- Degree of a field extension
- Fundamental theorem of symmetric polynomials
- The order of a finite field is a prime power
- The ring of integers
- Galois group of a polynomial
- Gaussian integers
- A polynomial over a field has at most as many roots as its degree
- 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
- Vieta's formulas
- Field generated by adjoining elements
- Algebraic closure
- Prime element
- Noetherian ring
- 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 Gaussian integers are a Euclidean domain
- The constructible numbers form a field closed under square roots
- Field
- The algebraic numbers form a field
- Gauss's lemma
- Euclidean domains are principal ideal domains
- 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
- Principal ideal domain
- 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
- Degree of a product
- 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
- Newton's identities
- Field extension
- Impossibility of squaring the circle
- Galois group
- Gauss–Wantzel theorem
- Primitive element of a finite field
- Hilbert's basis theorem
- Elementary symmetric polynomials
- K(a) is isomorphic to K[x] modulo the minimal polynomial
- Abel–Ruffini theorem
- Krull dimension
- Regular local rings are UFDs
- Associated prime
- Associative algebra
- Algebraic set
- Grothendieck's Galois theory
- Formally real field
- Primitive polynomial over a finite field
- Cohn's irreducibility criterion
- Nilradical
- Field of formal Laurent series
- Nilradical is the intersection of the prime ideals
- Complete intersection
- Galois group of a prime degree polynomial with two nonreal roots
- Jordan algebra
- Fredholm alternative
- Artin–Tate lemma
- Linear difference equation
- Artin–Schreier theorem
- Perron's irreducibility criterion
- Algebraic independence
- Lasker–Noether theorem
- Gröbner basis
- Characteristic of a field
- Normed field
- Finite Desarguesian planes are Pappian
- Field trace
- Splitting circle method
- Model theory of algebraically closed fields
- Purely transcendental extension
- Purely inseparable extension
- Radical extension
- Jacobson radical
- Symbolic power
- Weil conjectures
- Resultant
- Field norm
- Krull domain
- Non-Archimedean absolute value
- Finite field arithmetic
- Polynomial greatest common divisor
- Class number
- Height of a prime ideal
- Summation by parts
- Artinian reduced rings are semisimple
- Moduli space
- Lüroth's theorem
- Néron–Severi group
- Artin–Schreier theory
- Jacobson's conjecture
- Cyclotomic field
- Galois theory
- Cayley–Bacharach theorem
- Tropical variety
- Akizuki–Hopkins–Levitzki theorem
- Steinitz's theorem on algebraically closed fields
- Irreducible ideal
- Reduction to diagonal form by transvections
- Transcendence basis
- Newton polygon
- Iwasawa manifold
- Puiseux series
- Quasi-algebraically closed field
- Extension and contraction of ideals
- Standard conjectures on algebraic cycles
- Boundary integral method for the Dirichlet problem
- Coherent ring
- Absolute Galois group
- Function field
- Trace form
- Quantale
- Factorization of polynomials
- Polynomial interpolation
- Separably closed field
- Brauer group
- Noncommutative Gröbner basis
- Lagrange resolvent
- Homotopy continuation for polynomial systems
- Filtered algebra
- Complex p-adic numbers
- Prime avoidance lemma
- Krull–Akizuki theorem
- Polynomial long division
- Galois connection between subgroups and subfields
- Ideal quotient
- Krasner's lemma
- Pythagorean field
- Separable closure
- Nagata's criterion for factoriality
- Normal closure
- Topological field
- Jacobson radical of an Artinian ring is nilpotent
- Deformation theory
- Capelli's theorem
- Zariski's main theorem
- Structure theorem for Artinian rings
- Milnor K-theory
- Berlekamp's algorithm
- Transcendental extension
- Jump relations for layer potentials
- Mori–Nagata theorem
- Field homomorphism
- Ordered ring
- Conway polynomial
- Field of rational functions
- Berkovich space dynamics
- Embedded prime
- Real closure
- Bézout's theorem
- Reduced ring
- Primary decomposition
- Casus irreducibilis
- Kovacic's algorithm
- Galois group of a finite field extension
- Commuting matrices
- Perfect field
- Hilbert's irreducibility theorem
- Minimal prime ideal
- Subalgebra
- Veblen–Young theorem
- Infinite Galois theory
- GCD domain
- Transcendence degree
- Matrix congruence
- Characterizations of Galois extensions
- Differential graded algebra
- Commutative algebra over a ring
- Faithful module
- Cyclic extension
- Normal basis theorem
- Hilbertian field
- Separably generated extension
- Generic smoothness
- Lagrange interpolation
- Levitzki's theorem
- Lenagan's theorem
- Categorical theory
- Solvability of x^n - a
- Derivation
- Liouvillian function
- Mori domain
- Ring of invariants
- Atomic domain
- Differential Galois theory
- Quintic equation
- Zero divisors are the union of associated primes
- Homogeneous polynomial
- Tensor product of fields
- Shafarevich's theorem on solvable Galois groups
- Krull's principal ideal theorem
- Abelian extension
- Kummer–Artin–Schreier–Witt theory
- Valuation
- Linearly disjoint extensions
- Differential field
- Discriminant of a field extension
- Krull's height theorem
- Hahn series
- Cyclotomic extension
- Étale fundamental group
- Cohen's theorem on Noetherian rings
- Primary ideal
- Radical ideal
- Krull topology
- Perfect closure
- Tsen's theorem
- Galois' theorem on solvability
- Nyström method
- Annihilator
- Hopkins–Levitzki theorem
- Existence of transcendence bases
- Prüfer domain
- Impossibility of the classical construction problems
- Azumaya algebra
- Bézout domain
- Compositum of fields
- Artin's theorem on fixed fields
- Superalgebra
- AKS primality test
- Fredholm's theorem
- Finiteness of associated primes
- Separable degree
- Factorization of polynomials over finite fields
- Divisor class group
- Buchberger's algorithm
- Artin–Schreier theorem on formally real fields
- Hilbert's Theorem 90
- Inverse Galois problem
- Integer factorization
- Valuation ring
- Horner's method
- Hadamard product
- Solvable by radicals
- Radical of an ideal
- Artinian ring
- Newton–Puiseux theorem
- Discrete valuation
Further reading
About this page
| Tag | 0GYGHKX834 |
|---|---|
| URI | m2://0GYGHKX834 |
| Kind | definition |
| Writing | W1 stated |
| MSC | 13A99 |
| Defines | Commutative ring |
| Uses | Ring |
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What this page needs for W2
- edge not_instance_of in: one non-example that fails one clause (0 of 1)
The writing levels say how complete the text of a page is. They do not measure its truth.
What links here
Used by
- Ideal generated by a subset
- Freshman's dream
- Divisibility in a commutative ring
- Prime ideal
- Integral domain
- Polynomials in several variables
- Frobenius endomorphism
- The ring of integers
- Noetherian ring
- Field
- Krull dimension
- Associated prime
- Associative algebra
- Nilradical
- Nilradical is the intersection of the prime ideals
- Jacobson radical
- Symbolic power
- Resultant
- Height of a prime ideal
- Summation by parts
- Akizuki–Hopkins–Levitzki theorem
- Irreducible ideal
- Extension and contraction of ideals
- Coherent ring
- Quantale
- Filtered algebra
- Prime avoidance lemma
- Polynomial long division
- Ideal quotient
- Ordered ring
- Reduced ring
- Primary decomposition
- Commuting matrices
- Minimal prime ideal
- Differential graded algebra
- Commutative algebra over a ring
- Derivation
- Tensor product of fields
- Cohen's theorem on Noetherian rings
- Primary ideal
- Radical ideal
- Annihilator
- Azumaya algebra
- Superalgebra
- Horner's method
- Hadamard product
- Radical of an ideal
- Artinian ring