Lie ring

Definition
A Lie ring is a set $$L$$ equipped with the structure of an abelian group (operation denoted $$+$$) and a bracket operation $$[\ ,\ ]$$ (called a Lie bracket) satisfying the following additional conditions:

A Lie ring which is also an algebra over a field (or a commutative unital ring) is termed a Lie algebra over that field (or commutative unital ring).

If $$R$$ is a commutative unital ring and $$L$$ is additionally equipped with the structure of a $$R$$-module, and the Lie bracket of $$L$$ is $$R$$-bilinear, then this makes $$L$$ a $$R$$-Lie algebra.

Viewpoints

 * The Lie rings form a variety of algebras called the variety of Lie rings.
 * The Jacobi identity for Lie rings is a close counterpart to associativity in groups.
 * The associated Lie ring for a strongly central series and the associated Lie ring for a group form a convenient way to pass from groups to Lie rings.
 * The Malcev correspondence and its generalization, the Lazard correspondence, form convenient ways to move back and forth between groups and Lie rings without loss of information.

Related notions

 * Lie algebra is a variation where the underlying group is a module over a specified commutative unital ring and the Lie bracket is a bilinear map.
 * Multiplicative Lie ring is a generalization to cases where the underlying group is possibly non-abelian.