{'cats': '<div class="catlinks" data-mw="interface" id="catlinks"><div class="mw-normal-catlinks" id="mw-normal-catlinks"><a href="/index.php/Special:Categories" title="Special:Categories">Category</a>: <ul><li><a href="/index.php/Category:TeX_done" title="Category:TeX done">TeX done</a></li></ul></div></div>',
 'code': True,
 'name': '*-regular_ring',
 'text': '<div class="mw-content-ltr" dir="ltr" id="mw-content-text" lang="en"><p>A <a href="/index.php/Regular_ring_(in_the_sense_of_von_Neumann)" title="Regular ring (in the sense of von Neumann)">regular ring (in the sense of von Neumann)</a> admitting an involutory <a class="mw-redirect" href="/index.php/Anti-automorphism" title="Anti-automorphism">anti-automorphism</a> $\\alpha \\mapsto \\alpha^*$ such that $\\alpha \\alpha^* = 0$ implies $\\alpha=0$. An idempotent $e$ of a $*$-regular ring is called a <i>projector</i> if $e^* = e$. Every left (right) ideal of a $*$-regular ring is generated by a unique projector. One can thus speak of the lattice of projectors of a $*$-regular ring. If this lattice is <a href="/index.php/Complete_lattice" title="Complete lattice">complete</a>, then it is a continuous geometry. A complemented <a href="/index.php/Modular_lattice" title="Modular lattice">modular lattice</a> (cf. also <a href="/index.php/Lattice_with_complements" title="Lattice with complements">Lattice with complements</a>) having a homogeneous basis $a_1,\\ldots,a_n$, where $n \\ge 4$, is an ortho-complemented lattice if and only if it is isomorphic to the lattice of projectors of some $*$-regular ring.\n</p>\n<h4><span class="mw-headline" id="References">References</span></h4>\n<table>\n<tbody><tr><td valign="top">[1]</td> <td valign="top">  L.A. Skornyakov,   "Complemented modular lattices and regular rings" , Oliver &amp; Boyd  (1964)  (Translated from Russian)</td></tr>\n<tr><td valign="top">[2]</td> <td valign="top">  S.K. Berberian,   "Baer $*$-rings" , Springer  (1972)</td></tr>\n<tr><td valign="top">[3]</td> <td valign="top">  I. Kaplansky,   "Rings of operators" , Benjamin  (1968)</td></tr>\n</tbody></table>\n\n\n\n\n<div class="springerCitation" id="citeRevision"><strong>How to Cite This Entry:</strong><br/> *-regular ring. <i>Encyclopedia of Mathematics.</i> URL: http://www.encyclopediaofmath.org/index.php?title=*-regular_ring&amp;oldid=39400</div><div class="springerCitation" id="originalIsbn">This article was adapted from an original article by L.A. Skornyakov (originator),  which appeared in Encyclopedia of Mathematics - ISBN 1402006098. <a href="http://www.encyclopediaofmath.org/index.php?title=*-regular_ring&amp;oldid=15998">See original article</a></div></div>'}