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

{\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

{\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 $H(a,b)$  over a field $k$  is a division algebra defined by three generators $i,j,k$  with the relations

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

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

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

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

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

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

$*$  $i$  $j$  $k$
$i$  $a$  $-k$  $-aj$
$j$  $k$  $b$  $bi$
$k$  $aj$  $-bi$  $-ab$

### Commutative algebra

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

## 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.
• 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.