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Difference between revisions of "Centralizer"

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The subset of a ring, group or semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021220/c0212201.png" /> consisting of elements that commute (are interchangable) with all elements of a certain set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021220/c0212202.png" />; the centralizer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021220/c0212203.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021220/c0212204.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021220/c0212205.png" />. 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).
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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====
 
====References====

Revision as of 19:53, 11 April 2014

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)

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How to Cite This Entry:
Centralizer. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Centralizer&oldid=16843
This article was adapted from an original article by L.A. Bokut (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article