---
tag: 0B6QC0C0G1
kind: definition
name: Ring isomorphism
msc: [16B99]
defines: 0KN2S81K72
uses: [0GGE0Z0JQ1]
sources:
  - url: https://en.wikipedia.org/wiki/Ring_homomorphism
    locator: Endomorphisms, isomorphisms, and automorphisms
---

## Definition

Let $R$ and $S$ be [[rings|0894J980RK]].
A ring isomorphism from $R$ to $S$ is a [[ring homomorphism|0GGE0Z0JQ1]] $f \colon R \to S$ 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.