VeblenWedderburn system
A VeblenWedderburn system is an algebraic system over a set $R$ with two binary operations^{} $+$ (called addition) and $\cdot $ (called multiplication) on $R$ such that

1.
there is a $0\in R$, and that $R$ is an abelian group^{} under $+$, with $0$ the additive identity

2.
$R\{0\}$, together with $\cdot $, is a loop (we denote $1$ as its identity element^{})

3.
$\cdot $ is right distributive^{} over $+$; that is, $(a+b)\cdot c=a\cdot c+b\cdot c$

4.
if $a\ne b$, then the equation $x\cdot a=x\cdot b+c$ has a unique solution in $x$
A VeblenWedderburn system is also called a quasifield.
Usually, we write $ab$ instead of $a\cdot b$.
For any $a,b,c\in R$, by defining a ternary operation $*$ on $R$, given by
$$a*b*c:=ab+c,$$ 
it is not hard to see that $(R,*,0,1)$ is a ternary ring. In fact, it is a linear ternary ring because $ab=a*b*0$ and $a+c=a*1*c$.
For example, any field, or more generally, any division ring, associative or not, is VeblenWedderburn. An example of a VeblenWedderburn system that is not a division ring is the Hall quasifield.
A wellknown fact about VeblenWedderburn systems is that, the projective plane^{} of a VeblenWedderburn system is a translation plane, and, conversely, every translation plane can be coordinatized by a VeblenWedderburn system. This is the reason why a translation plane is also called a VeblenWedderburn plane.
Remark. Let $R$ be a VeblenWedderburn system. If the multiplication $\cdot $, in addition to be right distributive over $+$, is also left distributive over $+$, then $R$ is a semifield. If $\cdot $, on the other hand, is associative, then $R$ is an abelian nearfield (a nearfield such that $+$ is commutative^{}).
References
 1 R. Casse, Projective Geometry, An Introduction, Oxford University Press (2006)
Title  VeblenWedderburn system 
Canonical name  VeblenWedderburnSystem 
Date of creation  20130322 19:15:06 
Last modified on  20130322 19:15:06 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  9 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 51A35 
Classification  msc 51A40 
Classification  msc 51E15 
Classification  msc 51A25 
Synonym  VeblenWedderburn ring 
Synonym  quasifield 
Synonym  VW system 
Synonym  VW system 