# *-regular ring

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.

#### 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://www.encyclopediaofmath.org/index.php?title=*-regular_ring&oldid=39400