Nearring

Definition
A nearring or near-ring or near ring, short for right nearring or right near-ring or right near ring''', is a set $$N$$ equipped with the following operations:


 * A (infix) binary operation $$+$$
 * A unary operation $$-$$
 * A (infix) binary operation $$*$$
 * A constant $$0$$

such that:


 * 1) $$(N,+,0,-)$$ is a defining ingredient::group (not necessarily abelian): $$N$$ forms a group with multiplication $$+$$, identity element $$0$$, and inverse operation $$-$$.
 * 2) $$(N,*)$$ is a defining ingredient::semigroup under multiplication, i.e., $$*$$ is an associative binary operation.
 * 3) The following distributivity law holds: $$\! (x + y) * z = (x * z) + (y * z)$$

There is a corresponding notion of left nearring. By default, nearring means right nearring.