Baer ring

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 formula_3. Since this makes "R" isomorphic to its opposite ring "R"op, the definition of Rickart *-ring is left-right symmetric.
Properties.
The projections in a Rickart *-ring form a lattice, which is complete if the ring is a Baer *-ring.