# Fully invariant subgroup of additive group 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

## Definition

Let $L$ be a Lie ring. A subset $S$ of $L$ is termed a fully invariant subgroup of the additive group of $L$ if it satisfies the following equivalent conditions:

1. Consider the additive group of $L$. Then, $S$ is a fully invariant subgroup of the additive group of $L$.
2. $S$ is a Lie subring of $L$ that is also fully invariant as an additive subgroup of $L$.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
homomorph-containing subgroup of additive group of a Lie ring subset of a Lie ring that is a homomorph-containing subgroup of the additive group of the Lie ring. |FULL LIST, MORE INFO
verbal subgroup of additive group of a Lie ring subset of a Lie ring that is a verbal subgroup of the additive group of the Lie ring. |FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
fully invariant Lie subring invariant under all Lie ring endomorphisms fully invariant subgroup of additive group of Lie ring is derivation-invariant and fully invariant Fully invariant derivation-invariant Lie subring, Fully invariant ideal of a Lie ring, Lie subring invariant under any additive endomorphism satisfying a comultiplication condition|FULL LIST, MORE INFO
derivation-invariant Lie subring fully invariant subgroup of additive group of Lie ring is derivation-invariant and fully invariant Fully invariant derivation-invariant Lie subring, Lie subring invariant under any additive endomorphism satisfying a comultiplication condition|FULL LIST, MORE INFO
ideal of a Lie ring (via derivation-invariant) (via derivation-invariant) |FULL LIST, MORE INFO