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 explicitly, the opposite of a ring (R, +, ·) is the ring (R, +, ∗) whose multiplication ∗ is defined by a ∗ b = b · a. The opposite ring can be used to define multimodules, a generalization of bimodules. They also help clarify the relationship between left and right modules (see #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 R is isomorphic to R.
- A ring and its opposite ring are anti-isomorphic.
- A ring is commutative if and only if its operation coincides with its opposite operation.
- The left ideals of a ring are the right ideals of its opposite.
- The opposite ring of a field is a field (regardless of whether the field is commutative).
- A left module over a ring is a right module over its opposite, and vice versa.