Associative rings and algebras Definition
Ring isomorphism
Builds on Ring homomorphism
Definition
Let and be rings. A ring isomorphism from to is a ring homomorphism that is bijective. The rings and are isomorphic, written , if there is a ring isomorphism from to .
Remarks
The inverse map of a ring isomorphism is again a ring homomorphism. A ring isomorphism from to is an automorphism of . Complex conjugation is an automorphism of .
Reading path
To understand this page from the ground up, read these 4 pages in order. The foundations come first.
Show the path as a graph
- no evidence level
- Group concept
- Ring definition
- Ring homomorphism definition
- Ring isomorphism definition
Where this leads
50 pages build on this one.
- Uniqueness of finite fields of a given order
- Uniqueness of the algebraic closure
- Galois field
- Uniqueness of splitting fields
- Third isomorphism theorem for rings
- Order of the Galois group of a Galois extension equals its degree
- Impossibility of doubling the cube
- Impossibility of trisecting the angle
- Second isomorphism theorem for rings
- Galois group of a polynomial
- Fixed field
- Degree of a simple extension is the degree of the minimal polynomial
- The Galois group has order at most the degree
- The degree of a constructible number is a power of two
- Fundamental theorem of Galois theory
- Frobenius automorphism of a finite field
- Chinese remainder theorem for rings
- Impossibility of squaring the circle
- Galois group
- Gauss–Wantzel theorem
- First isomorphism theorem for rings
- K(a) is isomorphic to K[x] modulo the minimal polynomial
- Abel–Ruffini theorem
- Galois group of a prime degree polynomial with two nonreal roots
- Model theory of algebraically closed fields
- Weil conjectures
- Galois theory
- Steinitz's theorem on algebraically closed fields
- Standard conjectures on algebraic cycles
- Absolute Galois group
- Lagrange resolvent
- Galois connection between subgroups and subfields
- Berlekamp's algorithm
- Galois group of a finite field extension
- Infinite Galois theory
- Characterizations of Galois extensions
- Cyclic extension
- Normal basis theorem
- Categorical theory
- Solvability of x^n - a
- Quintic equation
- Shafarevich's theorem on solvable Galois groups
- Abelian extension
- Étale fundamental group
- Krull topology
- Galois' theorem on solvability
- Impossibility of the classical construction problems
- Artin's theorem on fixed fields
- Hilbert's Theorem 90
- Inverse Galois problem
Further reading
- Web https://en.wikipedia.org/wiki/Ring_homomorphism, Endomorphisms, isomorphisms, and automorphisms
About this page
| Tag | 0B6QC0C0G1 |
|---|---|
| URI | m2://0B6QC0C0G1 |
| Kind | definition |
| Writing | W0 stub |
| MSC | 16B99 |
| Defines | Ring isomorphism |
| Uses | Ring homomorphism |
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