Opposite ring

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.

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

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

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 ${\displaystyle R}$.