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 ab = b · a. Ring addition is per definition commutative.


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.


  1. Berrick & Keating (2000), p. 19


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