Opposite ring
In 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.[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 be viewed as categories with a single object. The construction of the opposite category generalizes the opposite group, opposite ring, etc.
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 isomorphic to 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.[3]
- The opposite ring of a field is a field (regardless of whether the field is commutative).[4]
- A left module over a ring is a right module over its opposite, and vice versa.[5]
Notes
- ↑ Berrick & Keating (2000), p. 19
- 1 2 Bourbaki 1989, p. 101.
- ↑ Bourbaki 1989, p. 103.
- ↑ Bourbaki 1989, p. 114.
- ↑ Bourbaki 1989, p. 192.
References
See also
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