Baer ring

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In mathematics, 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.

1 Definitions
2 Examples
3 Properties
4 References

[edit] Definitions

Given a subset S of a ring R, the left annihilator of S is the set {r ε R : rS = 0}.
An idempotent in a ring is an element p with p² = p.
A projection in a *-ring is an idempotent p that is self adjoint (p*=p).
A (left) Rickart ring is a ring such that the left annihilator of any element is generated (as a left ideal) by an idempotent element.
A Rickart *-ring is a *-ring such that left annihilator of any element is generated (as a left ideal) by a projection.
A (left) Baer ring (named after Reinhold Baer) is a ring such that the left annihilator of any subset is generated (as a left ideal) by an idempotent element.
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 Kaplansky) is a C* algebra that is also a Baer *-ring.

[edit] Examples

von Neumann algebras are examples of all the different sorts of ring above.

[edit] Properties

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

[edit] References

Baer ring on PlanetMath
L.A. Skornyakov (2001), “Regular ring (in the sense of von Neumann)”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
L.A. Skornyakov (2001), “Rickart ring”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
J.D.M. Wright (2001), “AW* algebra”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
Sterling K Berberian Baer *-rings ISBN 0-387-05751-X
I. Kaplansky, Rings of Operators, W. A. Benjamin, Inc., New York, 1968.
C.E. Rickart, Banach algebras with an adjoint operation Ann. of Math. , 47 (1946) pp. 528–550