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Encyclopaedia of Mathematics

Part of the book series: Encyclopaedia of Mathematics ((ENMA,volume 7))

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Abstract

*-Regular Ring - A regular ring (in the sense of von Neumann) admitting an involutory antiautomorphism α→α such that αα * = 0 implies α = 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..., a n , where n≥4, is an ortho-complemented lattice if and only if it is isomorphic to the lattice of projectors of some *-regular ring.

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Hazewinkel, M. (1991). R. In: Hazewinkel, M. (eds) Encyclopaedia of Mathematics. Encyclopaedia of Mathematics, vol 7. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-1237-4_4

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