---
tag: M8Q24XB0AF
kind: concept
name: Opposite ring
msc: [16B99]
sources:
  - url: https://en.wikipedia.org/wiki/Opposite_ring
same_as:
  wikidata: Q1940482
  wikipedia: Opposite ring
---

## Summary

The opposite ring of a [[ring|0B1261B5ED]] $R$ has the same elements and the same addition as $R$, with the factors of every product swapped.
A ring equals its opposite exactly when it is commutative.
The opposite ring turns right modules into left modules and back.
So results proved for one side carry over to the other.
It also appears in the endomorphism ring of a ring viewed as a module over itself.

## Explanations

### L1 human

Take a ring and read every product backwards: the new product of $a$ and $b$ is the old product $ba$.
The result is again a ring, called the opposite ring.
For commutative rings nothing changes.
For matrices over a commutative ring such as $\R$, the transpose turns $AB$ into $B^T A^T$, so the transpose map identifies the matrix ring with its opposite.