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.)
A ring (R, +, · ) is commutative if, and only if, its opposite is commutative. If two rings R1 and R2 are isomorphic, then their corresponding opposite rings are also 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.
- Berrick & Keating (2000),
- Berrick, A. Jon; Keating, M. E.. An introduction to rings and modules with K-theory in view. Volume 65 of Cambridge studies in advanced mathematics. Cambridge University Press, 2000. ISBN 978-0-521-63274-4
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