Fastring

In mathematics, a fast ring is the generalization of the algebraic structure of a ring in which the addition no longer has to be commutative and in which only a one-sided distributive law applies. In general, fast rings are used to be able to work algebraically with functions on groups.

Definitions

Fastring

A right fast ring or fast ring for short is an algebraic structure with two two- digit combinations of addition and multiplication for which the following applies: ${\displaystyle (F,+,\cdot )}$ ${\displaystyle +}$ ${\displaystyle \cdot }$

1. ${\displaystyle (F,+)\,}$is a group .
2. ${\displaystyle (F,\cdot )}$is a half group .
3. The right side distributive is valid: for all ${\displaystyle (a+b)\cdot c=(a\cdot c)+(b\cdot c)}$ ${\displaystyle a,b,c\in F.}$

${\displaystyle (F,+,\cdot )}$on the other hand, a left fastring is called if instead of the right-hand distributive law

  3. ′ the distributive law on the left is valid: for all ${\displaystyle c\cdot (a+b)=(c\cdot a)+(c\cdot b)}$ ${\displaystyle a,b,c\in F.}$

If a fast ring fulfills both distributive laws , it is called a distributive fast ring , i.e. a right and left fast ring .

A fast ring in which the additive group is commutative is called Abelian . However, if the multiplicative semigroup is commutative, then on the other hand is known as commutative . Commutative fast rings are always distributive. ${\displaystyle (F,+,\cdot )}$ ${\displaystyle (F,+)\,}$ ${\displaystyle (F,\cdot )}$ ${\displaystyle (F,+,\cdot )}$

For the sake of simplicity, products are also written without the multiplication sign for all and to save in brackets, as usual, in the following, the multiplication is always stronger than the addition. ${\displaystyle ab:=a\cdot b}$ ${\displaystyle a,b\in F}$

If you define a two-digit link subtraction according to ${\displaystyle (F,+,\cdot )}$ ${\displaystyle -}$

${\displaystyle a-b:=a+(-b)\,}$ for all ${\displaystyle a,b\in F,}$

so also applies to this because of

${\displaystyle (a-b)c+bc=(a-b+b)c=(a+0)c+0=ac-(bc)+bc\,}$

the right-sided distributive law: for all ${\displaystyle (a-b)\cdot c=(a\cdot c)-(b\cdot c)}$ ${\displaystyle a,b,c\in F.}$

The corresponding left-hand distributive law of subtraction applies analogously to a left fast ring.

Zero element

According to the definition, every fast ring has a neutral element 0 with respect to the addition, i.e. H. ${\displaystyle (F,+,\cdot )}$

${\displaystyle 0+a=a+0=a\,}$ for all ${\displaystyle a\in F.}$

This is called the zero element or, for short, the zero of the right or left fast ring. In a (right) fast ring it is left absorbing with regard to the multiplication :

${\displaystyle 0\cdot a=(a-a)a=aa-aa=0}$ for all ${\displaystyle a\in F}$

and in the case of a left fastring it is absorbent to the right , but the zero is generally not absorbent on both sides.

One element

If a fast ring also has a neutral element 1 with regard to the multiplication, ${\displaystyle (F,+,\cdot )}$

${\displaystyle 1\cdot a=a\cdot 1=a}$ for all ${\displaystyle a\in F,}$

so this is called the one element or, for short, the one of the fast ring.

Almost body

→ Main article : almost body

If you also form a group, the fast ring is called the fast body . It can be shown that the additive group is then Abelian . ${\displaystyle (F\setminus \{0\},\cdot )}$ ${\displaystyle (F,+,\cdot )}$

Semi-fast ring

Every fast ring can be generalized to a half fast ring , in which the definition of the fast ring only requires addition instead of the group property: ${\displaystyle (F,+,\cdot )}$

      1. ′    is a half group. ${\displaystyle (F,+)\,}$

Examples

• Typical examples of fast rings are sets of self-mapping to groups. For example, be a group and denote the set of all functions , then the group structure is carried over to ${\displaystyle (G,+)\,}$ ${\displaystyle G^{G}}$ ${\displaystyle f\colon G\to G}$ ${\displaystyle G^{G}}$

${\displaystyle (f+g)(x):=f(x)+g(x)\,}$ for all ${\displaystyle x\in G.}$

In addition, the composition forms a monoid , so that there is then a fast ring with one , since the right-hand distributive law is automatically fulfilled: ${\displaystyle G^{G}}$ ${\displaystyle \circ }$ ${\displaystyle \left(G^{G},+,\circ \right)}$ ${\displaystyle \operatorname {id} _{G}}$

${\displaystyle ((f+g)\circ h)(x)=(f+g)\left(h(x)\right)=f\left(h(x)\right)+g\left(h(x)\right)=(f\circ h)(x)+(g\circ h)(x)=(f\circ h+g\circ h)(x)}$ for all ${\displaystyle x\in G.}$

• Is a group and a subgroup of the automorphism group of that operates sharply transitive on , i.e. H. for two elements there is exactly one with , then you can define an operation on as follows : You choose a fixed element . Are , so there are unique elements with and . You then define , and you set for all . Then there is an almost body whose multiplicative group is isomorphic to . The right-hand distributivity law is therefore fulfilled for everyone . If , the automorphism group of contains a subgroup which is isomorphic to the quaternion group of order 8. This group operates sharply transitive on . This gives a minimal example of an almost body that is not a body. ${\displaystyle \ (F,+,0)}$ ${\displaystyle \ A}$ ${\displaystyle \ F}$ ${\displaystyle F\setminus \{0\}}$ ${\displaystyle x,y\in F\setminus \{0\}}$ ${\displaystyle g\in A}$ ${\displaystyle \ g(x)=y}$ ${\displaystyle \ \circ }$ ${\displaystyle F}$ ${\displaystyle e\in F\setminus \{0\}}$ ${\displaystyle \ a,b\in F\setminus \{0\}}$ ${\displaystyle \ g,h\in A}$ ${\displaystyle \ g(e)=a}$ ${\displaystyle \ h(e)=b}$ ${\displaystyle \ a\circ b:=g(h(e))}$ ${\displaystyle \ 0\circ a=a\circ 0=0}$ ${\displaystyle \ a\in F}$ ${\displaystyle \ (F,+,\circ ,0,e)}$ ${\displaystyle \ A}$ ${\displaystyle g(h(e)+k(e))=g(h(e))+g(k(e))}$ ${\displaystyle g,h,k\in A}$ ${\displaystyle \ F=Z_{3}^{2}}$ ${\displaystyle F}$ ${\displaystyle F\setminus \{0\}}$

properties

• Every fast ring has a 0-symmetrical part and a constant part so that applies. ${\displaystyle (F,+,\cdot )}$ ${\displaystyle F_{0}=\{a\in F\mid a\cdot 0=0\}}$ ${\displaystyle F_{c}=\{a\in F\mid a\cdot 0=a\},}$ ${\displaystyle F_{0}\cap F_{c}=\{0\}}$

See also

• Body (algebra)
• Oblique body
• Half ring (algebraic structure)

Remarks

1. ↑ does not have to be commutative ! ${\displaystyle (F,+)}$

literature

• James R. Clay: Nearrings. Geneses and applications. Oxford University Press, Oxford 1992, ISBN 0-19-853398-5 .
• John DP Meldrum: Near-rings and their links with groups. Pitman, Boston 1985, ISBN 0-273-08701-0 .
• Günter Pilz: Near-Rings. North-Holland, Amsterdam-New York-Oxford 1977, ISBN 0-7204-0566-1 (Rev. ed. 1983).
• Heinz Wähling: Theory of almost bodies. Thales Verlag, 1987.