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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
, the opposite of a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
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
bimodule In abstract algebra, a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. Besides appearing naturally in many parts of mathematics, bimodules play a clarifying role, in t ...
s. They also help clarify the relationship between left and right
modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
(see ').
Monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...
s,
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
, rings, and
algebras In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...
can all be viewed as
categories Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) * Categories (Peirce) * ...
with a single
object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ...
. The construction of the
opposite category In category theory, a branch of mathematics, the opposite category or dual category ''C''op of a given category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields t ...
generalizes the
opposite group In group theory, a branch of mathematics, an opposite group is a way to construct a group from another group that allows one to define right action as a special case of left action. Monoids, groups, rings, and algebras can be viewed as ca ...
, 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 confusion with some unary operation.
A ring R having isomorphic opposite ring is called a ''self-opposite'' ring, which name indicates that R^\text is essentially the same as R.
All commutative rings are self-opposite.
Let us define the antiisomorphism : \iota\colon (R,\diamond) \to (R,\cdot), where \iota(a) = a for a \in R. It is indeed an antiisomorphism, since \iota(a\diamond b) = a \diamond b = b\cdot a = \iota(b)\cdot\iota(a). The antiisomorphism \iota 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: \Rightarrow: Let (R,\cdot) be self-opposite. If f\colon (R,\cdot) \to (R,\diamond) is an isomorphism, then \iota\circ f, being a composition of antiisomorphism and isomorphism, is an antiisomorphism from (R,\cdot) to itself, hence antiautomorphism.
\Leftarrow: If g\colon(R,\cdot)\to(R,\cdot) is an antiautomorphism, then (\iota^\circ g)\colon(R,\cdot)\to(R,\diamond) is an isomorphism as a composition of two antiisomorphisms. So (R,\cdot) is self-opposite. and If (R,\cdot) is self-opposite and the group of automorphisms \operatorname(R,\cdot) 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 q, then the map h\mapsto q\circ h, where h runs over \operatorname(R,\cdot), is clearly injective but also surjective, since each antiautomorphism g = q \circ(q^\circ g) = q\circ h for some automorphism h.
It can be proven in a similar way, that under the same assumptions the number of isomorphisms from (R,\cdot) to (R,\diamond) equals the number of antiautomorphisms of (R,\cdot). If some antiautomorphism g is also an automorphism, then for each a, b\in(R,\cdot) :g(a \cdot b) = g(b) \cdot g(a) = g(b \cdot a) Since g is bijective, a \cdot b = b \cdot a for all a and b, 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.
Denote by G the group of all automorphisms together with all antiautomorphisms and set n = \operatorname\operatorname(R). The above remarks imply, that \operatornameG = 2n if a ring (or rng) is noncommutative and self-opposite. If it is commutative or non-self-opposite, then \operatornameG = n.


Examples


The smallest noncommutative ring with unity

The smallest such ring R 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 C_2\times C_2\times C_2 = C_2^3. Obviously R^\text is antiisomorphic to R, as is always the case, but it is also isomorphic to R. Below are the tables of addition and multiplication in R, and multiplication in the opposite ring, which is a transposed table. To prove that the two rings are isomorphic, take a map f: R^\text \to R given by the table The map swaps elements in only two pairs: 3 \leftrightarrow 4 and 5 \leftrightarrow 7. Rename accordingly the elements in the multiplication table for \diamond (arguments and values). Next, rearrange rows and columns to bring the arguments back to ascending order. The table becomes exactly the multiplication table of R. Similar changes in the table of additive group yield the same table, so f is an automorphism of this group, and since f(1)=1, it is indeed a ring isomorphism.
The map is involutory, i.e. f \circ f = \text, so f^=f and it is an isomorphism from R to R^\text equally well.
So, the permutation f can be reinterpreted to define isomorphism f\colon(R,\cdot)\rightarrow(R,\diamond) and then q = \iota\circ f is an antiautomorphism of (R,\cdot) given by the same permutation q = (3,4)(5,7).
The ring R has exactly two automorphisms: identity \operatorname_R and p=(3,5)(4,7), that is \operatorname(R) = \. So its full group G has four elements with two of them antiautomorphisms. One is q and the second, denote it by r, can be calculated :r = q\circ p = (3,4)(5,7)(3,5)(4,7) = (3,7)(4,5) :G = \ = \ There is no element of order 4, so the group is not cyclic and must be the group D_2 (or the Klein group K_4), 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 2 x 2 matrices over the field with 3 elements \text_3 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. The largest noncommutative ring S 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 S is used. Two things should be kept in mind: that the element denoted by 18 is the unity of S and that 1 is not the unity. The additive group of S is C_3^3. The group of all automorphisms \operatorname(S) has 3 elements: :\ \operatorname_S :h_1 = (1,13,25)(2,26,14)(4,16,19)(5,20,17)(7,10,22)(8,23,11) :h_2 = (1,25,13)(2,14,26)(4,19,16)(5,17,20)(7,22,10)(8,11,23) = h_1^ Since S is self-opposite, it has also 3 antiautomorphisms. One isomorphism f\colon(S,\cdot)\to(S,\diamond) is f = (1,14,13,2,25,26)(4,20,16,17,19,5)(7,8,10,23,22,11)(3,15)(6,21)(12,24),
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 R = (R,\cdot). The result is the multiplication table of R^\operatorname = (R,\diamond) and the additon table remains unchanged. Thus, one antiautomorphism :q_1 = \iota\circ f = (1,14,13,2,25,26)(4,20,16,17,19,5)(7,8,10,23,22,11)(3,15)(6,21)(12,24) is given by the same permutation. The other two can be calculated (in the multiplicative notation the composition symbol \circ can be dropped): \begin q_2 &= q_1h_1 = (1,14,13,2,25,26)(4,20,16,17,19,5)(7,8,10,23,22,11)(3,15)(6,21)(12,24)\cdot\\ &\quad \cdot(1,13,25)(2,26,14)(4,16,19)(5,20,17)(7,10,22)(8,23,11)\\ &= 1,14,13,2,25,26)(1,13,25)(2,26,14) 4,20,16,17,19,5)(4,16,19)(5,20,17)cdot\\ &\quad \cdot 7,8,10,23,22,11)(7,10,22)(8,23,11)(3,15)(6,21)(12,24)\\ &= (1,2)(13,26)(14,25)(4,17)(5,16)(19,20)(7,23)(8,22)(10,11)(3,15)(6,21)(12,24)\\ q_3 &= q_1h_2 = (1,26,25,2,13,14)(4,5,19,17,16,20)(7,11,22,23,10,8)(3,15)(6,21)(12,24) = q_1^ \end
Since q_1^2 = h_1, the group G is cyclic. If we set q=q_1, then
G = \.
The even powers are automorphisms and the odd ones – antiautomorphisms.


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.
One pair of rings R_1 and R_2 = R_1^\text has the additive group C_4 \times C_2 \times C_2 and the other pair R_3 and R_4 = R_3^\text, the group C_2 \times C_2 \times C_2 \times C_2 = C_2^4. 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 R_3^\text = R_4, they are opposite, but not isomorphic. The same is true for the pair R_1 and R_2, however, the ring \widetilde_2 listed in "The Book of the Rings" is not equal but only isomorphic to R_2.
The remaining 13−4=9 noncommutative rings are self-opposite.


Free algebra with two generators

The
free algebra In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting variables. Likewise, the po ...
k \langle x,y \rangle over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
k with generators x, y has multiplication from the multiplication of words. For example, :\begin (2x^2yx + 3yxy)\cdot(xyxy + 1) =& \text 2x^2yx^2yxy + 2x^2yx \\ & + 3yxyxyxy + 3yxy \end Then the opposite algebra has multiplication given by :\begin (2x^2yx + 3yxy)*(xyxy + 1) &= (xyxy + 1)\cdot (2x^2yx + 3yxy)\\ &= 2xyxyx^2yx + 3xyxy^2xy + 2x^2yx + 3yxy \end which are not equal elements.


Quaternion algebra

The quaternion algebra H(a,b) over a field F with a,b \in F^\times is a
division algebra In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. Definitions Formally, we start with a non-zero algebra ''D'' over a fie ...
defined by three generators i, j, k with the relations :i^2 = a,\ j^2 = b,\ k = ij = -ji All elements x \in H(a,b) are of the form :x = x_0 + x_ii + x_jj + x_kk, where x_0, x_i, x_j, x_k \in F For example, if F = \R, then H(-1,-1) is the usual quaternion algebra. If the multiplication of H(a,b) is denoted \cdot, it has the multiplication table : Then the opposite algebra H(a,b)^\text with multiplication denoted * has the table :


Commutative ring

A commutative ring (R,\cdot) is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to its opposite ring (R,*) = R^\text since x \cdot y = y \cdot x = x * y for all x and y in R. They are even equal (R,*)=(R,\cdot), since their operations are equal, i.e. * = \cdot.


Properties

* Two rings ''R''1 and ''R''2 are
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
if and only if their corresponding opposite rings are isomorphic. * The opposite of the opposite of a ring is identical with , that is ''R''opop = . * A ring and its opposite ring are anti-isomorphic. * A ring is
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
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 In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
is a division ring. * A left module over a ring is a right module over its opposite, and vice versa.


Notes


Citations


References

* *


See also

*
Opposite group In group theory, a branch of mathematics, an opposite group is a way to construct a group from another group that allows one to define right action as a special case of left action. Monoids, groups, rings, and algebras can be viewed as ca ...
*
Opposite category In category theory, a branch of mathematics, the opposite category or dual category ''C''op of a given category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields t ...
Ring theory {{Abstract-algebra-stub