# 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 ${\displaystyle k\langle x,y\rangle }$  over a field ${\displaystyle k}$  with generators ${\displaystyle x,y}$  has multiplication from the multiplication of words. For example,

{\displaystyle {\begin{aligned}(2x^{2}yx+3yxy)\cdot (xyxy+1)=&{\text{ }}2x^{2}yx^{2}yxy+2x^{2}yx\\&+3yxyxyxy+3yxy\end{aligned}}}

Then the opposite algebra has multiplication given by

{\displaystyle {\begin{aligned}(2x^{2}yx+3yxy)*(xyxy+1)&=(xyxy+1)\cdot (2x^{2}yx+3yxy)\\&=2xyxyx^{2}yx+3xyxy^{2}xy+2x^{2}yx+3yxy\end{aligned}}}

which are not equal elements.

### Quaternion algebra

The quaternion algebra ${\displaystyle H(a,b)}$ [3] over a field ${\displaystyle k}$  is a division algebra defined by three generators ${\displaystyle i,j,k}$  with the relations

${\displaystyle i^{2}=a}$ , ${\displaystyle j^{2}=b}$ , and ${\displaystyle k=ij=-ji}$

All elements of ${\displaystyle x\in H(a,b)}$  are of the form

${\displaystyle x=x_{0}+x_{i}i+x_{j}j+x_{k}k}$

If the multiplication of ${\displaystyle H(a,b)}$  is denoted ${\displaystyle \cdot }$ , it has the multiplication table

${\displaystyle \cdot }$  ${\displaystyle i}$  ${\displaystyle j}$  ${\displaystyle k}$
${\displaystyle i}$  ${\displaystyle a}$  ${\displaystyle k}$  ${\displaystyle aj}$
${\displaystyle j}$  ${\displaystyle -k}$  ${\displaystyle b}$  ${\displaystyle -bi}$
${\displaystyle k}$  ${\displaystyle -aj}$  ${\displaystyle bi}$  ${\displaystyle -ab}$

Then the opposite algebra ${\displaystyle H(a,b)^{\text{op}}}$  with multiplication denoted ${\displaystyle *}$  has the table

${\displaystyle *}$  ${\displaystyle i}$  ${\displaystyle j}$  ${\displaystyle k}$
${\displaystyle i}$  ${\displaystyle a}$  ${\displaystyle -k}$  ${\displaystyle -aj}$
${\displaystyle j}$  ${\displaystyle k}$  ${\displaystyle b}$  ${\displaystyle bi}$
${\displaystyle k}$  ${\displaystyle aj}$  ${\displaystyle -bi}$  ${\displaystyle -ab}$

### Commutative algebra

A commutative algebra ${\displaystyle (R,\cdot )}$  is isomorphic to its opposite algebra ${\displaystyle (R,*)=R^{\text{op}}}$  since ${\displaystyle x\cdot y=y\cdot x=x*y}$  for all ${\displaystyle x}$  and ${\displaystyle y}$  in ${\displaystyle R}$ .

## 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. ^ 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.