Near-ring

A near-ring is in mathematics, the generalization of the algebraic structure of a ring, in which the addition does not have to be commutative and in which only a one-sided distributive law. In general, near-rings are used to work with algebraic functions on groups.

Definitions

Fast Ring

A right near-ring or short fast ring is an algebraic structure with two two -digit addition and multiplication shortcuts for the following applies:

On the other hand is called, if, instead of the right side, a left distributive near-ring

3 ' the left- distributive law is valid: for all

Where a fast ring both distributive laws, as it is called distributive near-ring, so it is right and left near-ring.

This is called a near-ring in which the additive group is commutative, abelian. However, if the multiplicative semigroup is commutative, then one calls the other hand, as commutative. Almost Commutative rings are always distributive.

Products are written in a simplified form without the multiplication sign for all and saving tie clip as usual in the following multiplication always stronger than the addition.

If we define according to a near-ring a binary operation subtraction

As also applies to those for

Analogously, for a left near-ring the corresponding left- distributive law of subtraction.

Zero element

Each ring has almost according to the definition, a neutral element 0 with respect to addition, ie

This is called the zero element or just the zero of the right or left Almost ring. It is at a (legal) Fast Ring respect to multiplication on the left absorbing:

And a left near-ring quite absorbing, but the zero is not generally on both sides absorbing.

One element

Has a fast ring also a neutral element 1 with respect to multiplication,

Then we call this the unity element or just the one of the fast ring.

Nearly body

Also forms a group, called the Fast Ring Nearly body. It can be shown that the additive group is then abelian.

Almost half ring

Each near-ring can be generalized even to a semi- near-ring in which the definition of fast ring in place of the group property of the addition is only to be requested:

1 ' is a semigroup.

Examples

• Typical examples of near-rings are sets of self-maps on groups. Be about a group and denote the set of all functions, then the group structure transmits on by

• Is a group and a subgroup of the automorphism group of which operates sharply - transitive, ie for two elements

There is exactly one with, then you can define as follows an operation on: One chooses a fixed element. If so, there are certain elements and with clearly. One defines then, also are employed for all. Then a fast body whose multiplicative group is isomorphic. The right-side Distributivitätsgesetz is satisfied because for all. If so contains the automorphism group of a subgroup that is isomorphic to the quaternion group of order 8. This group operates on sharp - transitive. The result is a minimum example for a fast body which is not a body.

Properties

• Almost every ring has a 0 -symmetric part and a constant part that is considered so.