# Centralizer

From Encyclopedia of Mathematics

The subset of a ring, group or semi-group $R$ consisting of elements that commute (are interchangable) with all elements of a certain set $X\subseteq R$; the centralizer of $S$ in $R$ is denoted by $C_R(S)$. The centralizer of an irreducible subring (that is, one not stabilizing proper subgroups) of endomorphisms of an Abelian group in the ring of all endomorphisms of this group is a division ring (Schur's lemma).

#### References

[1] | N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956) |

**How to Cite This Entry:**

Centralizer.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Centralizer&oldid=31550

This article was adapted from an original article by L.A. Bokut (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article