Counter ring

The counter ring to a ring is a construction from the mathematical branch of ring theory . The counter-ring to a ring is created by swapping the factors during multiplication.

definition

Let it be a ring. Then the opposite ring is defined as follows: ${\displaystyle R}$ ${\displaystyle R^{op}}$

• The underlying set of is . ${\displaystyle R^{op}}$ ${\displaystyle R}$
• The addition + on is the same as that on . ${\displaystyle R^{op}}$ ${\displaystyle R}$
• The multiplication is defined by the multiplication of as follows: for all . ${\displaystyle \circ }$ ${\displaystyle \cdot }$ ${\displaystyle R}$ ${\displaystyle a\circ b:=b\cdot a}$ ${\displaystyle a,b\in R^{op}}$

${\displaystyle R^{op}}$ is essentially the output ring, only in the case of multiplication the order of the factors is reversed compared to the output ring.

properties

• Is commutative, then is obvious . ${\displaystyle R}$ ${\displaystyle R^{op}=R}$
• Sentences about left ideals in a ring are sentences about right ideals in . Therefore theorems that apply to all left ideals in all rings also apply to right ideals in all rings. ${\displaystyle R}$ ${\displaystyle R^{op}}$
• If an -algebra is over a field , then there is also such an algebra by using the same vector space structure for and . One then speaks of counter-algebra. ${\displaystyle R}$ ${\displaystyle K}$ ${\displaystyle R^{op}}$ ${\displaystyle R}$ ${\displaystyle R^{op}}$
• Let it be the algebra of - matrices over a field. Then, as is well known, the rule applies to transposition . This means that the transposition is a ring homomorphism , even an isomorphism . More generally, an antihomomorphism between two rings is a homomorphism or ${\displaystyle \mathrm {Mat} _{n}(K)}$ ${\displaystyle n\times n}$ ${\displaystyle A\mapsto A^{T}}$ ${\displaystyle (A\cdot B)^{T}=B^{T}\cdot A^{T}}$ ${\displaystyle \mathrm {Mat} _{n}(K)\rightarrow \mathrm {Mat} _{n}(K)^{op}}$ ${\displaystyle R\rightarrow S}$ ${\displaystyle R\rightarrow S^{op}}$ ${\displaystyle R^{op}\rightarrow S}$
• In general, and are not isomorphic. Examples are found where certain left-right symmetries do not apply. For example, there are left- noetherian rings that are not right-noetheric; such rings cannot be isomorphic to their mating ring. ${\displaystyle R}$ ${\displaystyle R^{op}}$
• Is a -Linksmodul so is by defining a -Rechtsmodul. ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle m\cdot a:=am,\,a\in R,m\in M}$ ${\displaystyle R^{op}}$

Individual evidence

1. ↑ Theodor Bröcker: Lineare Algebra and Analytical Geometry , Birkhäuser Verlag (2004), ISBN 3-0348-8962-3 , Chapter X, §8, page 331
2. ↑ Louis H. Rowen: Ring Theory. Volume 1. Academic Press Inc., Boston et al. 1988, ISBN 0-125-99841-4 ( Pure and Applied Mathematics 127), definition 0.1.11