Associative rings and algebras Definition
Subring
Builds on Ring
Definition
Let be a ring. A subring of is a subset with these properties.
- .
- If , then .
- If , then .
A subring is a ring with the addition and the multiplication of , restricted to .
Remarks
The integers are a subring of , and is a subring of . The set of even integers is closed under addition and multiplication, but it is not a subring of , because . Authors who do not require a ring to have an identity also do not require .
Reading path
To understand this page from the ground up, read these 3 pages in order. The foundations come first.
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Where this leads
126 pages build on this one.
- Transcendental element
- Uniqueness of finite fields of a given order
- Existence of splitting fields
- Uniqueness of the algebraic closure
- Intermediate field
- Splitting field
- Norm on the Gaussian integers
- Galois field
- Uniqueness of splitting fields
- The minimal polynomial divides every polynomial vanishing at the element
- Units of the Gaussian integers
- Subfield
- Subfields of a finite field
- Algebraic extension
- Tower law
- Order of the Galois group of a Galois extension equals its degree
- Normal extension
- A finite field of order q is the splitting field of x^q - x
- Impossibility of doubling the cube
- Finite extensions are algebraic
- Galois extension
- Existence of an algebraic closure
- Impossibility of trisecting the angle
- Transcendence of pi
- Minimal polynomial of an algebraic element
- Degree of a field extension
- The order of a finite field is a prime power
- Second isomorphism theorem for rings
- Galois group of a polynomial
- Gaussian integers
- Fixed field
- Field generated by adjoining elements
- Algebraic closure
- Separable extension
- Degree of a simple extension is the degree of the minimal polynomial
- The minimal polynomial is irreducible
- Algebraic element
- Finite extension
- Simple extension
- The Gaussian integers are a Euclidean domain
- The constructible numbers form a field closed under square roots
- The algebraic numbers form a field
- The Galois group has order at most the degree
- 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
- Prime subfield
- Field extension
- Impossibility of squaring the circle
- Galois group
- Gauss–Wantzel theorem
- K(a) is isomorphic to K[x] modulo the minimal polynomial
- Abel–Ruffini theorem
- Grothendieck's Galois theory
- Primitive polynomial over a finite field
- Galois group of a prime degree polynomial with two nonreal roots
- Artin–Schreier theorem
- Algebraic independence
- Field trace
- Model theory of algebraically closed fields
- Purely transcendental extension
- Purely inseparable extension
- Radical extension
- Weil conjectures
- Resultant
- Field norm
- Class number
- Lüroth's theorem
- Artin–Schreier theory
- Galois theory
- Steinitz's theorem on algebraically closed fields
- Center of a ring
- Transcendence basis
- Iwasawa manifold
- Standard conjectures on algebraic cycles
- Absolute Galois group
- Function field
- Trace form
- Separably closed field
- Lagrange resolvent
- Complex p-adic numbers
- Galois connection between subgroups and subfields
- Krasner's lemma
- Separable closure
- Normal closure
- Transcendental extension
- Berkovich space dynamics
- Cartan–Brauer–Hua theorem
- Real closure
- Casus irreducibilis
- Kovacic's algorithm
- Galois group of a finite field extension
- Subalgebra
- Infinite Galois theory
- Transcendence degree
- Characterizations of Galois extensions
- Cyclic extension
- Normal basis theorem
- Separably generated extension
- Categorical theory
- Solvability of x^n - a
- Liouvillian function
- Differential Galois theory
- Quintic equation
- Tensor product of fields
- Shafarevich's theorem on solvable Galois groups
- Abelian extension
- Linearly disjoint extensions
- Differential field
- Discriminant of a field extension
- Cyclotomic extension
- Étale fundamental group
- Krull topology
- Perfect closure
- Galois' theorem on solvability
- Existence of transcendence bases
- Impossibility of the classical construction problems
- Compositum of fields
- Artin's theorem on fixed fields
- Separable degree
- Hilbert's Theorem 90
- Inverse Galois problem
- Solvable by radicals
- Newton–Puiseux theorem
Further reading
- Web https://en.wikipedia.org/wiki/Subring, Definition
About this page
| Tag | 0859D58WPD |
|---|---|
| URI | m2://0859D58WPD |
| Kind | definition |
| Writing | W1 stated |
| MSC | 16B99 |
| Defines | Subring |
| 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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