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.
They also help clarify the relationship between left and right modules (see § Properties).
Monoids, groups, rings, and algebras can all be viewed as categories with a single object.
In this section the symbol for multiplication in the opposite ring is changed from asterisk to diamond, to avoid confusing it with some unary operations.
can be defined generally for semigroups, monoids, groups, rings, rngs, algebras.
In case of rings (and rngs) we obtain the general equivalence.
Proof: By the assumption and the above equivalence there exist antiautomorphisms.
It can be proven in a similar way, that under the same assumptions the number of isomorphisms from
By contraposition, if a ring is noncommutative (and self-opposite), then no antiautomorphism is an automorphism.
if a ring (or rng) is noncommutative and self-opposite.
,[d] and multiplication in the opposite ring, which is a transposed table.
To prove that the two rings are isomorphic, take a map
given by the table The map swaps elements in only two pairs:
Rename accordingly the elements in the multiplication table for
Next, rearrange rows and columns to bring the arguments back to ascending order.
The ring of the upper triangular 2 × 2 matrices over the field with 3 elements
listed in the "Book of the Rings" has 27 elements, and is also isomorphic.
In this section the notation from "The Book" for the elements of
, which can be verified using the tables of operations in "The Book" like in the first example by renaming and rearranging.
This time the changes should be made in the original tables of operations of
and the addition table remains unchanged.
The other five can be calculated (in the multiplicative notation the composition symbol
has 7 elements of order 2 (3 automorphisms and 4 antiautomorphisms) and can be identified as the dihedral group
All the rings with unity of orders ranging from 9 up to 15 are commutative,[5] so they are self-opposite.
[6] They can be coupled in two pairs of rings opposite to each other in a pair, and necessarily with the same additive group, since an antiisomorphism of rings is an isomorphism of their additive groups.
[3]: 433 Their tables of operations are not presented in this article, as they can be found in the source cited, and it can be verified that
[3]: 335 listed in "The Book of the Rings" is not equal but only isomorphic to
The remaining 13 − 4 = 9 noncommutative rings are self-opposite.
For example, Then the opposite algebra has multiplication given by which are not equal elements.
, it has the multiplication table Then the opposite algebra