Baer ring

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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.[1]
  • 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.[2] For unital rings, replacing all occurrences of 'left' with 'right' yields an equivalent definition, that is to say, the definition is left-right symmetric.[3]

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.

  • 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.

Examples

Properties

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

Notes

  1. Rickart rings are named after Template:Harvtxt who studied a similar property in operator algebras. This "principal implies projective" condition is the reason Rickart rings are sometimes called PP-rings. Template:Harv
  2. This condition was studied by Template:Harvs.
  3. T.Y. Lam (1999), "Lectures on Modules and Rings" ISBN 0-387-98428-3 pp.260

References

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