Abelian Lie ring
This article defines a Lie ring property: a property that can be evaluated to true/false for any Lie ring.
View a complete list of properties of Lie rings
VIEW RELATED: Lie ring property implications | Lie ring property non-implications |Lie ring metaproperty satisfactions | Lie ring metaproperty dissatisfactions | Lie ring property satisfactions | Lie ring property dissatisfactions
ANALOGY: This is an analogue in Lie ring of a property encountered in group. Specifically, it is a Lie ring property analogous to the group property: abelian group
View other analogues of abelian group | View other analogues in Lie rings of group properties (OR, View as a tabulated list)
An abelian Lie ring is a Lie ring satisfying the following equivalent conditions:
- The Lie bracket of any two elements is zero.
- Every Lie subring of the Lie ring is an ideal in the Lie ring.
Equivalence of definitions
Further information: Lie ring is abelian iff every subring is an ideal
Ring whose commutator operation is the Lie bracket
Suppose is an associative ring. can be viewed as a Lie ring with the Lie bracket as . The Lie ring is an abelian Lie ring if and only if is a commutative ring.
Group via the Lazard correspondence
Moreover, under the natural bijection from to , abelian subrings of correspond to abelian subgroups of .