Wikipedia / Opposite ring
Wikipedia / 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.
Relation to automorphisms and antiautomorphisms
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.
A ring is called a self-opposite ring if it is isomorphic to its opposite ring, is essentially the same as .
All commutative rings are self-opposite.
Let us define the antiisomorphism
for .
It is indeed an antiisomorphism, since can be defined generally for semigroups, monoids, groups, rings, rngs, algebras. In case of rings (and rngs) we obtain the general equivalence.
A ring is self-opposite if and only if it has at least one antiautomorphism.
Proof: is an isomorphism, then to itself, hence antiautomorphism.
is self-opposite.
and
is finite, then the number of antiautomorphisms equals the number of automorphisms.
Proof: By the assumption and the above equivalence there exist antiautomorphisms. If we pick one of them and denote it by , then the map runs over for some automorphism .
equals the number of antiautomorphisms of .
and , so the ring is commutative and all antiautomorphisms are automorphisms. 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. If it is commutative or non-self-opposite, then .
Examples
The smallest noncommutative ring with unity
has eight elements and it is the only noncommutative ring among 11 rings with unity of order 8, up to isomorphism. It has the additive group . is antiisomorphic to , as is always the case, but it is also isomorphic to . Below are the tables of addition and multiplication in , and multiplication in the opposite ring, which is a transposed table. given by the table and (arguments and values). Next, rearrange rows and columns to bring the arguments back to ascending order. The table becomes exactly the multiplication table of , it is indeed a ring isomorphism.
The map is involutory, i.e. equally well.
given by the same permutation .
, can be calculated
(the Klein group ), which can be confirmed by calculation. The "symmetry group" of this ring is isomorphic to the symmetry group of rectangle.
Noncommutative ring with 27 elements
The ring of the upper triangular has 27 elements and is a noncommutative ring. It is unique up to isomorphism, that is, all noncommutative rings with unity and 27 elements are isomorphic to it. is not the unity. is .
has 6 elements:
is and the addition table remains unchanged. Thus, one antiautomorphism
can be dropped):
has 7 elements of order 2 (3 automorphisms and 4 antiautomorphisms) and can be identified as the dihedral group (see List of small groups isomorphic to the symmetry group of 3-antiprism, in Schoenflies notation in short Hermann–Mauguin notation for 3-dimensional space.
The smallest non-self-opposite rings with unity
All the rings with unity of orders ranging from 9 up to 15 are commutative, so they are self-opposite. The rings, that are not self-opposite, appear for the first time among the rings of order 16. There are 4 different non-self-opposite rings out of the total number of 50 rings with unity having 16 elements (37 commutative and 13 noncommutative). 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.
and , the group . and listed in "The Book of the Rings" is not equal but only isomorphic to .
The remaining noncommutative rings are self-opposite.
Free algebra with two generators
The free algebra over a field has multiplication from the multiplication of words. For example,
Then the opposite algebra has multiplication given by
which are not equal elements.
Quaternion algebra
is a division algebra with the relations
are of the form
is the usual quaternion algebra.
, it has the multiplication table
{| class="wikitable" style="text-align: center"
|}
has the table
{| class="wikitable" style="text-align: center"
|}
Commutative ring
is isomorphic in . They are even equal , since their operations are equal, i.e. .
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 is identical with , that is (Rop)op = .
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 division ring is a division ring.
A left module over a ring is a right module over its opposite, and vice versa.