Near-ring

A near-ring is an algebraic structure similar to a ring, but with weaker axioms. Specifically, a near-ring is a set N equipped with two binary operations, usually denoted by "+" and "⋅", satisfying the following axioms:

1. (N, +) is a group (not necessarily abelian).
2. (N, ⋅) is a semigroup.
3. a ⋅ (b + c) = a ⋅ b + a ⋅ c for all a, b, c ∈ N (left distributive law).

Notice that the group (N, +) is not required to be abelian, and only the left distributive law is required (the right distributive law is not assumed). The identity element of (N, +) is usually denoted by 0.

A 0-symmetric near-ring is a near-ring that further satisfies the property 0 ⋅ a = 0 for all a ∈ N. Some authors include this as part of the definition of a near-ring, while others require it explicitly.

A distributively generated near-ring is a near-ring N where the multiplicative semigroup (N, ⋅) is generated by a set of elements S such that each element of S distributes over addition on both sides (i.e., s ⋅ (a + b) = s ⋅ a + s ⋅ b and (a + b) ⋅ s = a ⋅ s + b ⋅ s for all a, b ∈ N and s ∈ S).

Notable differences between rings and near-rings:

• Addition is not necessarily commutative in a near-ring.
• The right distributive law is not required in a near-ring.
• Even if a near-ring has a multiplicative identity (usually denoted by 1), it is not required that a ⋅ 1 = a for all a ∈ N. If a ⋅ 1 = a for all a ∈ N, then 1 is called a right identity. If 1 ⋅ a = a for all a ∈ N, then 1 is called a left identity.
• The element 0 is not necessarily a two-sided annihilator in a near-ring (i.e., it is not necessarily true that a ⋅ 0 = 0 for all a ∈ N). However, it is always true that 0 ⋅ a = 0 in a 0-symmetric near-ring.

Near-rings arise naturally in the study of functions from a group to itself under pointwise addition and composition, particularly in the context of endomorphism near-rings. They also have applications in areas such as cryptography and coding theory.

Different types of near-rings are studied based on the properties of their additive and multiplicative structures, such as near-fields, planar near-rings, and boolean near-rings.