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 group (not necessarily abelian): $N$ forms a group with multiplication $+$, identity element $0$, and inverse operation $-$.
2. $(N,*)$ is a 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.