Definition:Centralizer

Centralizer of a Ring
Let $$S$$ be a subset of a ring $$\left({R, +, \circ}\right)$$.

The centralizer of $$S$$ in $$R$$ is defined as:

$$C_R \left({S}\right) = \left\{{x \in R: \forall s \in S: s \circ x = x \circ s}\right\}$$

That is, the centralizer of $$S$$ is the set of elements of $$R$$ which commute with all elements of $$S$$.