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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.
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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====
 
====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>
 
<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>

Revision as of 17:12, 9 October 2016

A regular ring (in the sense of von Neumann) admitting an involutory anti-automorphism such that implies . An idempotent of a -regular ring is called a projector if . 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 complete, then it is a continuous geometry. A complemented modular lattice (cf. also Lattice with complements) having a homogeneous basis , where , is an ortho-complemented lattice if and only if it is isomorphic to the lattice of projectors of some -regular ring.

References

[1] L.A. Skornyakov, "Complemented modular lattices and regular rings" , Oliver & Boyd (1964) (Translated from Russian)
[2] S.K. Berberian, "Baer -rings" , Springer (1972)
[3] I. Kaplansky, "Rings of operators" , Benjamin (1968)
How to Cite This Entry:
*-regular ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=*-regular_ring&oldid=39393
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article