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.

ExamplesEdit

Free algebra with two generatorsEdit

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 algebraEdit

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 algebraEdit

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

PropertiesEdit

  • 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]

CitationsEdit

  1. ^ Berrick & Keating (2000), p. 19
  2. ^ a b 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.

ReferencesEdit

See alsoEdit