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 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 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.