This article defines a non-associative ring property: a property that an be evaluated to true or false for any non-associative ring.
View other non-associative ring properties
A Lie ring is a set equipped with the structure of an abelian group (operation denoted ) and a bracket operation (called a Lie bracket) satisfying the following additional conditions:
|Condition name||Explicit identities (all variable letters are universally quantified over )|
|-bilinear, also known as biadditive, also known as left and right distributive|| Additive in left coordinate: |
Additive in right coordinate:
|alternating (hence skew-symmetric)|| Alternation: |
The second condition (skew symmetry) follows from the first (alternation); the reverse implication holds only if is 2-torsion-free.
Note also that skew symmetry means that we need assume only one of the two additivity identities and it implies the other.
|Jacobi identity|| Left-normed version: |
The two versions are equivalent by skew symmetry.
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).
- 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.