Сақинаның кері сақинасы

Mathematical concept
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 1=a ∗ b = b ⋅ a for all a, b 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.

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 and has the additive group and the other pair and 1= R 4 = R 3^\text{op} , the group 1= \mathrm{C} 2 \times \mathrm{C} 2 \times \mathrm{C} 2 \times \mathrm{C} 2 = \mathrm{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 , they are opposite, but not isomorphic. The same is true for the pair and 1= R 2 , however, the ring listed in "The Book of the Rings" is not equal but only isomorphic to 1= R 2 The remaining 1=13 − 4 = 9 noncommutative rings are self opposite.

A commutative ring is isomorphic to its opposite ring since for all and in 1= R They are even equal 1= (R,*)=(R,\cdot) , since their operations are equal, i. e. 1= * = \cdot .

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.