Moduloid

A moduloid is a mathematical structure that generalizes the concept of a ring, particularly in the context of non-commutative rings and non-associative operations. The term was introduced by Jean-Louis Loday in the 1980s as a way to study non-associative algebras and their homological properties.

A moduloid consists of an abelian group (M, +) and a binary operation * : M x M -> M that satisfies certain axioms. Specifically, the operation * must be distributive over the group operation +, meaning that for all a, b, and c in M, the following equations must hold:

a * (b + c) = (a * b) + (a * c)

(b + c) * a = (b * a) + (c * a)

These axioms ensure that the moduloid structure is compatible with the underlying abelian group structure. Moduloids can be used to study various types of non-associative algebras, such as Lie algebras and Jordan algebras, by considering their underlying moduloid structure.

In addition to their algebraic properties, moduloids have been studied in the context of homological algebra, where they provide a framework for understanding the cohomology of non-associative algebras. The study of moduloids has also led to the development of new techniques in non-commutative algebra and the theory of operads.