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 a ∗ b = 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 ${\displaystyle H(a,b)^{op}}$ 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 ${\displaystyle (R,*)=R^{op}}$ 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 field is a field (this also holds for skew-fields).[5]
• A left module over a ring is a right module over its opposite, and vice versa.[6]

Notes

1. Berrick & Keating (2000), p. 19
2. Bourbaki 1989, p. 101. sfn error: no target: CITEREFBourbaki1989 (help)
3. Milne. Class Field Theory. p. 120.
4. Bourbaki 1989, p. 103. sfn error: no target: CITEREFBourbaki1989 (help)
5. Bourbaki 1989, p. 114. sfn error: no target: CITEREFBourbaki1989 (help)
6. Bourbaki 1989, p. 192. sfn error: no target: CITEREFBourbaki1989 (help)

References

• Berrick, A. J.; Keating, M. E. (2000). An Introduction to Rings and Modules With K-theory in View. Cambridge studies in advanced mathematics. 65. Cambridge University Press. ISBN 978-0-521-63274-4.
• Nicolas, Bourbaki (1989). Algebra I. Berlin: Springer-Verlag. ISBN 978-3-540-64243-5. OCLC 18588156.

See also

• Opposite group
• Opposite category