# Opposite ring

In algebra, the **opposite** of a ring is another ring with the same elements and addition operation, but with the multiplication performed in the reverse order. More explicitly, the opposite of a ring (*R*, +, **·**) is the ring (*R*, +, ∗) whose multiplication ∗ is defined by *a* ∗ *b* = *b* **·** *a*.[1][2] The opposite ring can be used to define multimodules, a generalization of bimodules. They also help clarify the relationship between left and right modules (see #Properties).

Monoids, groups, rings, and algebras can be viewed as categories with a single object. The construction of the opposite category generalizes the opposite group, opposite ring, etc.

## Properties

- Two rings
*R*_{1}and*R*_{2}are isomorphic if and only if their corresponding opposite rings are isomorphic - The opposite of the opposite of a ring
`R`is isomorphic to`R`. - A ring and its opposite ring are anti-isomorphic.
- A ring is commutative if and only if its operation coincides with its opposite operation.[2]
- The left ideals of a ring are the right ideals of its opposite.[3]
- The opposite ring of a field is a field (regardless of whether the field is commutative).[4]
- A left module over a ring is a right module over its opposite, and vice versa.[5]

## Notes

- Berrick & Keating (2000), p. 19
- Bourbaki 1989, p. 101.
- Bourbaki 1989, p. 103.
- Bourbaki 1989, p. 114.
- Bourbaki 1989, p. 192.

## References

## See also

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