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 (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
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
All commutative rings are self-opposite.
Let us define the antiisomorphism
, where for .[lower-alpha 2]
It is indeed an antiisomorphism, since
A ring[lower-alpha 3] is self-opposite if and only if it has at least one antiautomorphism.
Proof:
and
If
Proof: By the assumption and the above equivalence there exist antiautomorphisms. If we pick one of them and denote it by
It can be proven in a similar way, that under the same assumptions the number of isomorphisms from
If some antiautomorphism
Since
Denote by
Examples
The smallest noncommutative ring with unity
The smallest such ring
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To prove that the two rings are isomorphic, take a map
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |
0 | 1 | 2 | 4 | 3 | 7 | 6 | 5 |
The map swaps elements in only two pairs:
The map is involutory, i.e.
So, the permutation
The ring
There is no element of order 4, so the group is not cyclic and must be the group
Noncommutative ring with 27 elements
The ring of the upper triangular 2 x 2 matrices over the field with 3 elements
Since
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
is given by the same permutation. The other five can be calculated (in the multiplicative notation the composition symbol
The group
The smallest non-self-opposite rings with unity
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
The remaining 13−4=9 noncommutative rings are self-opposite.
Free algebra with two generators
The free algebra
Then the opposite algebra has multiplication given by
which are not equal elements.
Quaternion algebra
The quaternion algebra
All elements
, where
For example, if
If the multiplication of
Then the opposite algebra
Commutative ring
A commutative ring
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 R is identical with R, that is Ropop = 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
- ↑ 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
is the identity function on the set , it is not the identity as a morphism, since and are two different objects (if is noncommutative) and the identity morphism can be only from an object to itself. Therefore cannot be denoted as , when is understood as an abbreviation of . If is commutative, then and . - ↑ 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
- ↑ Berrick & Keating (2000), p. 19
- ↑ Jump up to: 2.0 2.1 Bourbaki 1989, p. 101.
- ↑ Jump up to: 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 Nöbauer, Christof (23 October 2000). "The Book of the Rings". https://web.archive.org/web/20070824132439if_/http://www.algebra.uni-linz.ac.at/~noebsi/pub/rings.ps.
- ↑ Jump up to: 4.0 4.1 4.2 Nöbauer, Christof (26 October 2000). "The Book of the Rings, Part II". http://www.algebra.uni-linz.ac.at/~noebsi/pub/ringsII.ps.
- ↑ Jump up to: 5.0 5.1 5.2 5.3 Sloane, N. J. A., ed. "Sequence A127708 (Number of non-commutative rings with 1)". OEIS Foundation. https://oeis.org/A127708.
- ↑ Jump up to: 6.0 6.1 Nöbauer, Christof (5 April 2002). "Numbers of rings on groups of prime power order". http://www.algebra.uni-linz.ac.at/~noebsi/ringtable.html.
- ↑ Sloane, N. J. A., ed. "Sequence A037291 (Number of rings with 1)". OEIS Foundation. https://oeis.org/A037291.
- ↑ Sloane, N. J. A., ed. "Sequence A127707 (Number of commutative rings with 1)". OEIS Foundation. https://oeis.org/A127707.
- ↑ Milne. Class Field Theory. pp. 120.
- ↑ Bourbaki 1989, p. 103.
- ↑ Bourbaki 1989, p. 114.
- ↑ Bourbaki 1989, p. 192.
References
- Berrick, A. J.; Keating, M. E. (2000). An Introduction to Rings and Modules With K-theory in View. Cambridge studies in advanced mathematics. 65. Cambridge University Press. ISBN 978-0-521-63274-4. http://www.cambridge.org/us/academic/subjects/mathematics/algebra/introduction-rings-and-modules-k-theory-view.
- Nicolas, Bourbaki (1989). Algebra I. Berlin: Springer-Verlag. ISBN 978-3-540-64243-5. OCLC 18588156.
See also
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