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.

Definitions

  1. the left annihilator of any single element of R is generated by an idempotent element.
  2. the left annihilator of any element is a direct summand of R.
  3. All principal left ideals are projective R modules.
  1. The left annihilator of any subset of R is generated by an idempotent element.
  2. 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. Since this makes R isomorphic to its opposite ring Rop, the definition of Rickart *-ring is left-right symmetric.
The projections in a Rickart *-ring form a lattice, which is complete if the ring is a Baer *-ring.