View source for Quasi-regular ring

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

<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>

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]]].

====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>