Ring isomorphism

Ring isomorphism
Tag	`0B6QC0C0G1`
URI	`m2://0B6QC0C0G1`
Kind	definition
Writing	W0 stub
MSC	16B99
Defines	Ring isomorphism
Uses	Ring homomorphism

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Definition

Let $R$ and $S$ be rings. A ring isomorphism from $R$ to $S$ is a ring homomorphism $f \colon R \to S$ that is bijective. The rings $R$ and $S$ are isomorphic, written $R \cong S$ , if there is a ring isomorphism from $R$ to $S$ .

Remarks

The inverse map of a ring isomorphism is again a ring homomorphism. A ring isomorphism from $R$ to $R$ is an automorphism of $R$ . Complex conjugation is an automorphism of $\mathbb{C}$ .

Explanations

L1 human

Two isomorphic rings are the same ring with other names for the elements. The isomorphism is the dictionary between the names.

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What this needs

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Group concept
Ring definition
Ring homomorphism definition
Ring isomorphism definition

What depends on this

22 nodes use this node, directly or through other nodes.

Uniqueness of finite fields of a given order 010PKCQPXP
Uniqueness of the algebraic closure 01VQ7C4FTB
Galois field 04SGPNT8R9
Uniqueness of splitting fields 069A3BC3EP
Third isomorphism theorem for rings 073C80EC63
Order of the Galois group of a Galois extension equals its degree 0BDJBH8B4Z
Impossibility of doubling the cube 0CQ357M0QS
Impossibility of trisecting the angle 0EWXPZCSZ0
Second isomorphism theorem for rings 0HJ4DTW6FN
Galois group of a polynomial 0JZY3R908H
Fixed field 0KMJ6919WS
Degree of a simple extension is the degree of the minimal polynomial 0N5QW9K7MH
The Galois group has order at most the degree 0QM98DNWVW
The degree of a constructible number is a power of two 0SSSM1D475
Fundamental theorem of Galois theory 0T1Q8D0FW8
Frobenius automorphism of a finite field 0VP4TBVJ19
Chinese remainder theorem for rings 0W97J8QED7
Impossibility of squaring the circle 0X7CVDE3BG
Galois group 0XVR72M51M
Gauss–Wantzel theorem 0Y34JMJNE9
First isomorphism theorem for rings 0ZMYX7GMH0
K(a) is isomorphic to K[x] modulo the minimal polynomial 0ZN39QDTP9

Sources

Web https://en.wikipedia.org/wiki/Ring_homomorphism, Endomorphisms, isomorphisms, and automorphisms

What links here

Used by

Third isomorphism theorem for rings 073C80EC63
Second isomorphism theorem for rings 0HJ4DTW6FN
Fixed field 0KMJ6919WS
Frobenius automorphism of a finite field 0VP4TBVJ19
Chinese remainder theorem for rings 0W97J8QED7
Galois group 0XVR72M51M
First isomorphism theorem for rings 0ZMYX7GMH0