# 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.^{[1]}

More precisely, the opposite of a ring (*R*, +, **·**) is the ring (*R*, +, ∗) whose multiplication ∗ is defined by *a* ∗ *b* = *b* **·** *a*. Ring addition is per definition commutative.

## 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 is isomorphic to that ring. A ring and its opposite ring are anti-isomorphic.

A commutative ring is always equal to its opposite ring. A non-commutative ring may or may not be isomorphic to its opposite ring.

## Notes

## References

- Berrick, A. J.; Keating, M. E. (2000).
*An Introduction to Rings and Modules With K-theory in View*. Cambridge studies in advanced mathematics.**65**. Cambridge University Press. ISBN 978-0-521-63274-4.

This article is issued from Wikipedia - version of the 9/13/2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.