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 a * b = b · a. (Ring addition is per definition always commutative.)

Properties

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.

Notes

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