View source for *-regular ring

A [[Regular ring (in the sense of von Neumann)|regular ring (in the sense of von Neumann)]] admitting an involutory [[anti-automorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077000/r0770002.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077000/r0770003.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077000/r0770004.png" />. An idempotent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077000/r0770005.png" /> of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077000/r0770006.png" />-regular ring is called a projector if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077000/r0770007.png" />. Every left (right) ideal of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077000/r0770008.png" />-regular ring is generated by a unique projector. One can thus speak of the lattice of projectors of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077000/r0770009.png" />-regular ring. If this lattice is complete, then it is a continuous geometry. A complemented [[Modular lattice|modular lattice]] (cf. also [[Lattice with complements|Lattice with complements]]) having a homogeneous basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077000/r07700010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077000/r07700011.png" />, is an ortho-complemented lattice if and only if it is isomorphic to the lattice of projectors of some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077000/r07700012.png" />-regular ring.

====References====
<table><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><TR><TD valign="top">[2]</TD> <TD valign="top">  S.K. Berberian,   "Baer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077000/r07700013.png" />-rings" , Springer  (1972)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I. Kaplansky,   "Rings of operators" , Benjamin  (1968)</TD></TR></table>