## 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.