Nearrings

A near-ring is an algebraic structure that generalizes the notion of a ring by relaxing one of its distributive laws. It consists of a set N equipped with two binary operations: addition (+) and multiplication (·). The pair (N, +) forms a group (the additive structure may be non-abelian in some definitions), and (N, ·) forms a semigroup, i.e., multiplication is associative. In addition, at least one distributive law holds: left distributivity, meaning a · (x + y) = (a · x) + (a · y) for all a, x, y in N. Some authors also consider right near-rings, where (x + y) · a = (x · a) + (y · a) for all a, x, y. A structure that satisfies both one-sided distributivities is often called a two-sided near-ring; if the additive group is abelian and both distributivities hold, the structure is a ring.

Left (or right) near-rings typically require only one side of distributivity, which makes them useful as a framework for studying systems where linearity holds on one side but not the other. For each a in N, the map L_a: x ↦ a · x is an endomorphism of the additive group (N, +); this reflects the linear action encoded by left multiplication.

Relation to rings: every ring is a near-ring (it satisfies both distributive laws, with an abelian additive group). Near-rings thus generalize rings by allowing non-abelian additive structure and potentially only one distributive law. They appear in various areas of algebra and its applications, including transformation structures and the study of nonassociative generalizations.

See also: ring, near-field, near-domain, endomorphism.