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Difference between revisions of "Quasi-regular ring"

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A [[Ring|ring]] in which every element is quasi-regular. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076680/q0766801.png" /> of an alternative (in particular, associative) ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076680/q0766802.png" /> is called quasi-regular if there is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076680/q0766803.png" /> such that
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A [[Ring|ring]] in which every element is quasi-regular. An element $a$ of an alternative (in particular, associative) ring $R$ is called quasi-regular if there is an element $a'\in R$ such that
  
<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/q/q076/q076680/q0766804.png" /></td> </tr></table>
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$$a+a'+aa'=a+a'+a'a=0.$$
  
The element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076680/q0766805.png" /> is called the quasi-inverse of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076680/q0766806.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076680/q0766807.png" /> is a ring with identity 1, then an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076680/q0766808.png" /> is quasi-regular with quasi-inverse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076680/q0766809.png" /> if and only if the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076680/q07668010.png" /> is invertible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076680/q07668011.png" /> with inverse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076680/q07668012.png" />. Every [[Nilpotent element|nilpotent element]] is quasi-regular. In an associative ring the set of all quasi-regular elements forms a group with respect to the operation of cyclic composition: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076680/q07668013.png" />. An important example of a quasi-regular ring is the ring of (non-commutative) formal power series without constant terms. There exist simple associative quasi-regular rings [[#References|[2]]].
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The element $a'$ is called the quasi-inverse of $a$. If $R$ is a ring with identity 1, then an element $a\in R$ is quasi-regular with quasi-inverse $a'$ if and only if the element $1+a$ is invertible in $R$ with inverse $1+a'$. Every [[Nilpotent element|nilpotent element]] is quasi-regular. In an associative ring the set of all quasi-regular elements forms a group with respect to the operation of cyclic composition: $x\cdot y=x+y+xy$. An important example of a quasi-regular ring is the ring of (non-commutative) formal power series without constant terms. There exist simple associative quasi-regular rings [[#References|[2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Jacobson,  "Structure of rings" , Amer. Math. Soc.  (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Sasiada,  P.M. Cohn,  "An example of a simple radical ring"  ''J. of Algebra'' , '''5''' :  3  (1967)  pp. 373–377</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Jacobson,  "Structure of rings" , Amer. Math. Soc.  (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Sasiada,  P.M. Cohn,  "An example of a simple radical ring"  ''J. of Algebra'' , '''5''' :  3  (1967)  pp. 373–377</TD></TR></table>

Revision as of 09:48, 26 April 2014

A ring in which every element is quasi-regular. An element $a$ of an alternative (in particular, associative) ring $R$ is called quasi-regular if there is an element $a'\in R$ such that

$$a+a'+aa'=a+a'+a'a=0.$$

The element $a'$ is called the quasi-inverse of $a$. If $R$ is a ring with identity 1, then an element $a\in R$ is quasi-regular with quasi-inverse $a'$ if and only if the element $1+a$ is invertible in $R$ with inverse $1+a'$. Every nilpotent element is quasi-regular. In an associative ring the set of all quasi-regular elements forms a group with respect to the operation of cyclic composition: $x\cdot y=x+y+xy$. An important example of a quasi-regular ring is the ring of (non-commutative) formal power series without constant terms. There exist simple associative quasi-regular rings [2].

References

[1] N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)
[2] E. Sasiada, P.M. Cohn, "An example of a simple radical ring" J. of Algebra , 5 : 3 (1967) pp. 373–377
How to Cite This Entry:
Quasi-regular ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-regular_ring&oldid=31927
This article was adapted from an original article by I.P. Shestakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article