Alternative algebra

The term Alternative Body is a generalization of the algebraic concept of body of mathematics. In the definition of the alternative body is dispensed with the commutative and associative law for multiplication. Instead, it is demanded that the multiplication has the property of alternativeness.

Every division ring is an alternative body of each alternative body is both a left and a right quasi body. Alternatively finite bodies are always body. (→ See also: Moufangebene ).

An important application of the Alternative Body found in synthetic geometry. Ruth Moufang proved in 1933 that every Moufangebene is isomorphic to a projective coordinate plane via an alternative body.

Definitions

A lot with two links and is an alternative body if:

  • Is an abelian group whose neutral element is referred to as 0;
  • Is a group of quasi- neutral element is referred to as 1;
  • To link the alternativeness applies:
  • Apply both distributive laws: and.

Core of an alternative body

Each alternative body is a left and a right quasi body. You can Analogous to quasi bodies for each alternative body define its core:

This core is uniquely determined by the definition and complies with the links from the Alternative Body the axioms of a skew field. Alternatively, the body is then exactly a skew fields, if it matches its core. Note that the core in general need not be (in the sense of inclusion ) maximum skew field in the alternative body.

Properties

From alternativeness continues to follow the Flexibility Act

The two Alternativitäten and flexibility and the law are " cyclical " Laws: Are two of these laws, then it follows the third.

In an alternative body further the Moufang identities for multiplication apply:

And

Ruth Moufang 1934 showed that any three different elements a, b, c, an alternative form of the body, satisfy the relation, a swash body. This is an intensification of a set of Artin. The set of Artin states that any two distinct elements generate a skew field. The skew field thus generated are accurate then subsets of the core when the generating elements are in this core.

Each alternative body is both a left-and a right module over each contained in its core skew field, ie in particular on the core itself

Examples

  • The best known example of a "real " alternative body, so that is not a division ring, are the (real) octonions. The core of this alternative body is the field of real numbers. In addition, contains infinitely many isomorphic to the complex numbers body.
  • Each body and each general division ring is an example of an alternative body.
Moufang plane Projective plane Translation plane Emil Artin Digital Object Identifier Max August Zorn
52641
de