Ideal of a Lie ring
This article describes a Lie subring property: a property that can be evaluated for a subring of a Lie ring
View a complete list of such properties
VIEW RELATED: Lie subring property implications | Lie subring property non-implications | Lie subring metaproperty satisfactions | Lie subring metaproperty dissatisfactions | Lie subring property satisfactions |Lie subring property dissatisfactions
ANALOGY: This is an analogue in Lie ring of a property encountered in group. Specifically, it is a Lie subring property analogous to the subgroup property: normal subgroup
View other analogues of normal subgroup | View other analogues in Lie rings of subgroup properties (OR, View as a tabulated list)
A subset of a Lie ring is termed an ideal in if is additively a subgroup and whenever and .
Further information: Ideal of a Lie algebra
A Lie algebra is a Lie ring that is simultaneously (i.e., with the same operations) an algebra over a field. An ideal of a Lie algebra is an ideal of the underlying Lie ring that is also a linear subspace, i.e., it is closed under multiplication by scalars in the field.
Ring whose commutator operation is the Lie bracket
Suppose is an associative ring: an abelian group with a distributive associative multiplication. We can define the Lie ring associated with as with the same addition and wit hthe Lie bracket given by the commutator operation .
Then, the property of being an ideal of the Lie ring is equivalent to the property of being a Lie ideal in . Being a Lie ideal is weaker than being a two-sided ideal. It is incomparable with the property of being a left ideal or being a right ideal. Moreover, a subset that is both a left ideal and a Lie ideal is two-sided ideal. Similarly, a subset that is both a right ideal and a Lie ideal is a two-sided ideal.