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 (R, +, ⋅) is the ring (R, +, ∗) whose multiplication ∗ is defined by a ∗ b = b ⋅ a for all a, b in R.[1][2] 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 § Properties).
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
[edit]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,[3][4][a] which name indicates that is essentially the same as .
All commutative rings are self-opposite.
Let us define the antiisomorphism
- , where for .[b]
It is indeed an antiisomorphism, since . The antiisomorphism can be defined generally for semigroups, monoids, groups, rings, rngs, algebras. In case of rings (and rngs) we obtain the general equivalence.
A ring[c] is self-opposite if and only if it has at least one antiautomorphism.
Proof: : Let be self-opposite. If is an isomorphism, then , being a composition of antiisomorphism and isomorphism, is an antiisomorphism from to itself, hence antiautomorphism.
: If is an antiautomorphism, then is an isomorphism as a composition of two antiisomorphisms. So is self-opposite.
and
If is self-opposite and the group of automorphisms 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 , where runs over , is clearly injective but also surjective, since each antiautomorphism for some automorphism .
It can be proven in a similar way, that under the same assumptions the number of isomorphisms from to equals the number of antiautomorphisms of .
If some antiautomorphism is also an automorphism, then for each
Since is bijective, for all 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.
Denote by the group of all automorphisms together with all antiautomorphisms. The above remarks imply, that if a ring (or rng) is noncommutative and self-opposite. If it is commutative or non-self-opposite, then .
Examples
[edit]The smallest noncommutative ring with unity
[edit]The smallest such ring has eight elements and it is the only noncommutative ring among 11 rings with unity of order 8, up to isomorphism.[5] It has the additive group .[3]: 76 Obviously is antiisomorphic to , as is always the case, but it is also isomorphic to . Below are the tables of addition and multiplication in ,[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
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |
0 | 1 | 2 | 4 | 3 | 7 | 6 | 5 |
The map swaps elements in only two pairs: and . Rename accordingly the elements in the multiplication table for (arguments and values). Next, rearrange rows and columns to bring the arguments back to ascending order. The table becomes exactly the multiplication table of . Similar changes in the table of additive group yield the same table, so is an automorphism of this group, and since , it is indeed a ring isomorphism.
The map is involutory, i.e. , so = and it is an isomorphism from to equally well.
So, the permutation can be reinterpreted to define isomorphism and then is an antiautomorphism of given by the same permutation .
The ring has exactly two automorphisms: identity and , that is . So its full group has four elements with two of them antiautomorphisms. One is and the second, denote it by , can be calculated
There is no element of order 4, so the group is not cyclic and must be the group (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
[edit]The ring of the upper triangular 2 × 2 matrices over the field with 3 elements 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.[5][6] The largest noncommutative ring 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 is used. Two things should be kept in mind: that the element denoted by is the unity of and that is not the unity.[4]: 369 The additive group of is .[4]: 330 The group of all automorphisms has 6 elements:
Since is self-opposite, it has also 6 antiautomorphisms. One isomorphism is , 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 . The result is the multiplication table of and the addition table remains unchanged. Thus, one antiautomorphism
is given by the same permutation. The other five can be calculated (in the multiplicative notation the composition symbol can be dropped):
The group has 7 elements of order 2 (3 automorphisms and 4 antiautomorphisms) and can be identified as the dihedral group [e] (see List of small groups).
In geometric analogy the ring has the "symmetry group" isomorphic to the symmetry group of 3-antiprism,[f] which is the point group in Schoenflies notation or in short Hermann–Mauguin notation for 3-dimensional space.
The smallest non-self-opposite rings with unity
[edit]All the rings with unity of orders ranging from 9 up to 15 are commutative,[5] 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[7] having 16 elements (37[8] commutative and 13[5] noncommutative).[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.
One pair of rings [3]: 330 and has the additive group [3]: 262 and the other pair [3]: 535 and ,[3]: 541 the group .[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 , they are opposite, but not isomorphic. The same is true for the pair and , however, the ring [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.
Free algebra with two generators
[edit]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
[edit]The quaternion algebra [9] over a field with is a division algebra defined by three generators with the relations
All elements are of the form
- , where
For example, if , then is the usual quaternion algebra.
If the multiplication of is denoted , it has the multiplication table
Then the opposite algebra with multiplication denoted has the table
Commutative ring
[edit]A commutative ring is isomorphic to its opposite ring since for all and in . They are even equal , since their operations are equal, i.e. .
Properties
[edit]- 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 R is identical with R, that is (Rop)op = R.
- A ring and its opposite ring are anti-isomorphic.
- A ring is commutative if and only if its operation coincides with its opposite operation.[2]
- The left ideals of a ring are the right ideals of its opposite.[10]
- The opposite ring of a division ring is a division ring.[11]
- A left module over a ring is a right module over its opposite, and vice versa.[12]
Notes
[edit]- ^ The self-opposite rings in "The Book of the Rings" are labeled "self-converse", which is a different name, but the meaning is clear.
- ^ Although ι is the identity function on the set R, it is not the identity as a morphism, since (R, ⋅) and (R, ⋄) are two different objects (if R is noncommutative) and the identity morphism can be only from an object to itself. Therefore, ι cannot be denoted as idR, when R is understood as an abbreviation of (R, ⋄). If (R, ⋅) is commutative, then (R, ⋄) = (R, ⋅) and ι = id(R,⋅) = id(R,⋄) = idR.
- ^ In this equivalence (and in the next equality) the ring can be quite general i.e. with or without unity, noncommutative or commutative, finite or infinite.
- ^ The tables of operations differ from those in the source. They were modified in the following way. The unity 4 was renamed to 1 and 1 to 4 in the addition and multiplication table, and the rows and columns rearranged to position the unity 1 next to 0 for better clarity. Thus the two rings are isomorphic.
- ^ Symbol Dn is meant to abbreviate Dihn, the dihedral group with 2n elements, i.e. geometric convention is used.
- ^ The name 3-antiprism is here understood as the right 3-gonal antiprism that is not uniform, i.e. its side faces are not equilateral triangles. If they were equilateral, the antiprism would be the regular octahedron having the symmetry group larger than D3d.
Citations
[edit]- ^ Berrick & Keating (2000), p. 19
- ^ a b Bourbaki 1989, p. 101.
- ^ a b c d e f g h Nöbauer, Christof (23 October 2000). "The Book of the Rings".
- ^ a b c Nöbauer, Christof (26 October 2000). "The Book of the Rings, Part II". Archived from the original on 2007-08-24.
- ^ a b c d Sloane, N. J. A. (ed.). "Sequence A127708 (Number of non-commutative rings with 1)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ a b Nöbauer, Christof (5 April 2002). "Numbers of rings on groups of prime power order". Archived from the original on 2006-10-02.
- ^ Sloane, N. J. A. (ed.). "Sequence A037291 (Number of rings with 1)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A127707 (Number of commutative rings with 1)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Milne. Class Field Theory. p. 120.
- ^ Bourbaki 1989, p. 103.
- ^ Bourbaki 1989, p. 114.
- ^ Bourbaki 1989, p. 192.
References
[edit]- Berrick, A. J.; Keating, M. E. (2000). An Introduction to Rings and Modules With K-theory in View. Cambridge studies in advanced mathematics. Vol. 65. Cambridge University Press. ISBN 978-0-521-63274-4.
- Bourbaki, Nicolas (1989). Algebra I. Berlin: Springer-Verlag. ISBN 978-3-540-64243-5. OCLC 18588156.
See also
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