*-regular ring

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A regular ring (in the sense of von Neumann) admitting an involutory anti-automorphism $\alpha \mapsto \alpha^*$ such that $\alpha \alpha^* = 0$ implies $\alpha=0$. An idempotent $e$ of a $*$-regular ring is called a projector 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 complete, then it is a continuous geometry. A complemented modular lattice (cf. also Lattice with complements) 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.


[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:*-regular_ring&oldid=39400
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article