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 ab = 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.

Examples

Free algebra with two generators

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

The quaternion algebra [3] over a field is a division algebra defined by three generators with the relations

, , and

All elements of are of the form

If the multiplication of is denoted , it has the multiplication table

Then the opposite algebra with multiplication denoted has the table

Commutative algebra

A commutative algebra is isomorphic to its opposite algebra since for all and in .

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.[4]
  • The opposite ring of a division ring is a division ring.[5]
  • A left module over a ring is a right module over its opposite, and vice versa.[6]

Citations

  1. Berrick & Keating (2000), p. 19
  2. Bourbaki 1989, p. 101.
  3. Milne. Class Field Theory. p. 120.
  4. Bourbaki 1989, p. 103.
  5. Bourbaki 1989, p. 114.
  6. Bourbaki 1989, p. 192.

References

See also

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