*-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)