Near-field (mathematics)

In mathematics a near-field is a structure, very much like a division ring, except for one axiom : the right distributive law.

A near-field ${\displaystyle (Q,+,.)}$ is a structure, where + and . are binary operations (the respective addition and multiplication) on Q, satisfying these axioms :

• (${\displaystyle Q,+)}$ is an abelian group with identity element 0.
• ${\displaystyle (Q_{0},.)}$ is a group
• ${\displaystyle a.(b+c)=a.b+a.c\ \forall a,b,c\in Q}$

One can prove that the near-fields are just the quasifields with an associative multiplication.

We construct a near-field that is not a division ring of nine elements. Suppose K=GF(9). Let + and . denote the addition and multiplication respectively. We now define a new multiplication ${\displaystyle \otimes }$ on the same set K :

${\displaystyle u\otimes v=u.v}$ if u is square in the original field

${\displaystyle u\otimes v=u.v^{3}}$ if u is not square in the original field

One can check that ${\displaystyle (K,+,\otimes )}$ is a near-field but not a division ring.

This near-field allows the construction of a projective plane that is not Desarguesian: the Hall plane.

• Near-ring

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