Quadring

A quadring, sometimes also referred to as a quaternary ring, is a generalization of the concept of a ring in abstract algebra. Specifically, it refers to an algebraic structure that satisfies all the axioms of a ring except for associativity of multiplication. In other words, a quadring is a set Q equipped with two binary operations, typically denoted by + (addition) and ⋅ (multiplication), such that:

• (Q, +) is an abelian group (closed under addition, associative addition, existence of an additive identity, existence of additive inverses, and commutativity of addition).
• (Q, ⋅) is a magma (closed under multiplication).
• Multiplication distributes over addition; that is, for all a, b, c in Q:
  • a ⋅ (b + c) = (a ⋅ b) + (a ⋅ c) (left distributivity)
  • (b + c) ⋅ a = (b ⋅ a) + (c ⋅ a) (right distributivity)

The absence of associativity for multiplication is the defining characteristic that distinguishes a quadring from a ring. This relaxation of the ring axioms allows for the study of algebraic structures where the order of operations in a product matters.

The concept of quadrings is less commonly explored than rings, fields, or other more restrictive algebraic structures. Research in this area focuses on understanding the properties and implications of non-associative multiplication within a ring-like framework. It may arise in contexts where strict associativity is not a natural requirement, and the distributive laws are sufficient to capture relevant algebraic relationships. Studying quadrings can offer insights into the fundamental role of associativity in more familiar algebraic structures.