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 (see "").
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 formula_1 over a field formula_2 with generators formula_3 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 formula_6 over a field formula_2 is a division algebra defined by three generators formula_8 with the relations
All elements of formula_12 are of the form
If the multiplication of formula_6 is denoted formula_15, it has the multiplication table
Then the opposite algebra formula_16 with multiplication denoted formula_17 has the table
Commutative algebra.
A commutative algebra formula_18 is isomorphic to its opposite algebra formula_19 since formula_20 for all formula_21 and formula_22 in formula_23.