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 $k\langle x,y\rangle$ 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 $H(a,b)$ ^[3] 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 $H(a,b)^{op}$ 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 $(R,*)=R^{op}$ 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.^[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

^ Berrick & Keating (2000), p. 19
^ ^a ^b Bourbaki 1989, p. 101.
^ Milne. Class Field Theory. p. 120.
^ Bourbaki 1989, p. 103.
^ Bourbaki 1989, p. 114.
^ Bourbaki 1989, p. 192.

References

Berrick, A. J.; Keating, M. E. (2000). An Introduction to Rings and Modules With K-theory in View. Cambridge studies in advanced mathematics. Vol. 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.