In abstract algebra and functional analysis, Baer rings, *Baer -rings, Rickart rings, *Rickart -rings, and AW*-algebras are various attempts to give an algebraic analogue of von Neumann algebras, using axioms about annihilators of various sets. Any von Neumann algebra is a Baer *-ring, and much of the theory of projections in von Neumann algebras can be extended to all Baer *-rings, For example, Baer *-rings can be divided into types I, II, and III in the same way as von Neumann algebras. In the literature, left Rickart rings have also been termed left PP-rings. ("Principal implies projective": See definitions below.)

Definitions

In operator theory, the definitions are strengthened slightly by requiring the ring R to have an involution. Since this makes R isomorphic to its opposite ring Rop, the definition of Rickart *-ring is left-right symmetric.

Examples

Properties

The projections in a Rickart *-ring form a lattice, which is complete if the ring is a Baer *-ring.