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Difference between revisions of "Centre of a ring"

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The collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021270/c0212701.png" /> of all elements of the ring that commute with every element, that is,
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021270/c0212702.png" /></td> </tr></table>
 
  
The centre of a ring is a subring containing together with every invertible element its inverse. The centre of a ring that is an algebra with a unit element over a field contains the ground field (see [[Central algebra|Central algebra]]).
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The centre of a ring $R$ is
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the collection $Z$ of all elements of the ring $R$ that commute with every
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element, that is,
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$$Z=\{z: az = za \textrm{ for all }a \in R\}.$$
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The centre of a ring is a subring containing
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together with every invertible element its inverse. The centre of a
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ring that is an algebra with a unit element over a field contains the
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ground field (see
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[[Central algebra|Central algebra]]).

Latest revision as of 21:43, 5 March 2012

2020 Mathematics Subject Classification: Primary: 08-XX [MSN][ZBL]


The centre of a ring $R$ is the collection $Z$ of all elements of the ring $R$ that commute with every element, that is, $$Z=\{z: az = za \textrm{ for all }a \in R\}.$$ The centre of a ring is a subring containing together with every invertible element its inverse. The centre of a ring that is an algebra with a unit element over a field contains the ground field (see Central algebra).

How to Cite This Entry:
Centre of a ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Centre_of_a_ring&oldid=18869
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article