# 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

1. the left annihilator of any single element of R is generated (as a left ideal) by an idempotent element.
2. (For unital rings) the left annihilator of any element is a direct summand of R.
3. All principal left ideals (ideals of the form Rx) are projective R modules.
• A Baer ring has the following definitions:
1. The left annihilator of any subset of R is generated (as a left ideal) by an idempotent element.
2. (For unital rings) The left annihilator of any subset of R is a direct summand of R. For unital rings, replacing all occurrences of 'left' with 'right' yields an equivalent definition, that is to say, the definition is left-right symmetric.

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

• A projection in a *-ring is an idempotent p that is self adjoint (p*=p).
• A Rickart *-ring is a *-ring such that left annihilator of any element is generated (as a left ideal) by a projection.
• A Baer *-ring is a *-ring such that left annihilator of any subset is generated (as a left ideal) by a projection.
• An AW* algebra, introduced by Template:Harvtxt, is a C* algebra that is also a Baer *-ring.

## Properties

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