# Centralizer of a subset of a Lie ring

ANALOGY:This is an analogue in Lie rings of a term encountered in group. That term is: centralizer.

View other analogues of centralizer

## Definition

Let be a Lie ring and be a subset of . The **centralizer** of in , denoted , is defined as:

.

The centralizer of any subset of a Lie ring is a Lie subring. `For full proof, refer: Centralizer of subset of Lie ring is Lie subring`

## As a Galois correspondence

### Brief description

The centralizer operator can be viewed as a Galois correspondence from the collection of subsets of the Lie ring to itself, corresponding to the symmetric relation of *commuting* in the sense of the Lie bracket being zero. In particular:

- .
- .

### Implications

The implications of the above correspondence are as follows. Define the bicentralizer of a subset as the centralizer of its centralizer. Thus, a subset equals its own bicentralizer if and only if it occurs as the centralizer of some subset. A subset that occurs as a bicentralizer is termed a c-closed Lie subring. In particular, it is a subring. Also, the centralizer of any subset equals the centralizer of the Lie subring generated by that subset.