Opposite ring


In mathematics, specifically abstract 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 is the ring whose multiplication ∗ is defined by for all in R. The opposite ring can be used to define multimodules, a generalization of bimodules. They also help clarify the relationship between left and right modules.
Monoids, groups, rings, and algebras can all be viewed as categories with a single object. The construction of the opposite category generalizes the opposite group, opposite ring, etc.

Examples

Free algebra with two generators

The free algebra over a field with generators has multiplication from the multiplication of words. For example,
Then the opposite algebra has multiplication given by
which are not equal elements.

Quaternion algebra

The quaternion algebra over a field is a division algebra defined by three generators with the relations
All elements of are of the form
If the multiplication of is denoted, it has the multiplication table
Then the opposite algebra with multiplication denoted has the table

Commutative algebra

A commutative algebra is isomorphic to its opposite algebra since for all and in.

Properties