Opposite ring: Difference between revisions
removing a strange statement of something that is clearly stated later. (the ring and opposite ring are equal if the ring is commutative.) |
Wood Thrush (talk | contribs) No edit summary |
||
| Line 5: | Line 5: | ||
==Properties== |
==Properties== |
||
Two rings ''R''<sub>1</sub> and ''R''<sub>2</sub> are [[ring isomorphism|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. |
A commutative ring is always equal to its opposite ring. A [[non-commutative ring]] may or may not be isomorphic to its opposite ring. |
||
Revision as of 11:06, 19 June 2015
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 always commutative.)
Properties
Two rings R1 and R2 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. Vol. 65. Cambridge University Press. ISBN 978-0-521-63274-4.
 | This abstract algebra-related article is a stub. You can help Wikipedia by expanding it. |