## Axioms for Ring-like structures

Here are the explicit axioms for various algebraic
objects discussed.
A Ring is an algebra (R,+,*) such that
(R,+) is an abelian group, (R,*) is a semigroup, and the two distributive laws
apply, that is (a+b)*c = a*c + b*c and a*(b+c) = a*b + a*c.

Ring theory is somehow "linear", the existence of the
two distributive laws making the mappings induced by
multiplication, eg f(x) = a*x into endomorphisms of
the group: f(x+y) = a*(x+y) = a*x + a*y = f(x)+f(y).

Rings are often defined as the collection of
endomorphisms on
an abelian group under pointwise addition and
composition.
Any given ring is isomorphic to the ring of endomorphisms
induced by the multiplication on itself.

Near-rings, on the other hand, have only one distributive law.
A Near-ring is an algebra (N,+,*) such that
(N,+) is a group (not necessarily abelian),
(N,*) is a semigroup, and
one distributive law applies, eg
(a+b)*c=a*c + b*c in a right near-ring.

The induced functions are rarely endomorphisms.

Near-rings can be defined as a collection of
mappings on
a (not necessarily abelian) group, under the operations of pointwise
addition and composition.

A near-field is a rather more complex structure.
Axiomatically we say (N,+,*) is a near-field if
(N,+) is a group,
(N-{0}, *) is a group and
(a+b)*c = a*c + b*c, that is, the right distributive law holds.
Near-fields do not have simple interpretations as
collections of mappings as rings and near-rings have.