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.)
- Since the principal left ideals of a left hereditary ring or left semihereditary ring are projective, it is clear that both types are left Rickart rings. This includes von Neumann regular rings, which are left and right semihereditary. If a von Neumann regular ring R is also right or left self injective, then R is Baer.
- Any semisimple ring is Baer, since all left and right ideals are summands in R, including the annihilators.
- Any domain is Baer, since all annihilators are except for the annihilator of 0, which is R, and both and R are summands of R.
- The ring of bounded linear operators on a Hilbert space are a Baer ring and is also a Baer *-ring with the involution * given by the adjoint.
- von Neumann algebras are examples of all the different sorts of ring above.
The projections in a Rickart *-ring form a lattice, which is complete if the ring is a Baer *-ring.
Rickart rings are named after Rickart (1946) who studied a similar property in operator algebras. This "principal implies projective" condition is the reason Rickart rings are sometimes called PP-rings. (Lam 1999)
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- Kaplansky, I. (1968), Rings of Operators, New York: W. A. Benjamin, Inc.
- Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98428-5, MR 1653294
- Rickart, C. E. (1946), "Banach algebras with an adjoint operation", Annals of Mathematics, Second Series, 47 (3): 528–550, doi:10.2307/1969091, JSTOR 1969091, MR 0017474
- L.A. Skornyakov (2001) [1994], "Regular ring (in the sense of von Neumann)", Encyclopedia of Mathematics, EMS Press
- L.A. Skornyakov (2001) [1994], "Rickart ring", Encyclopedia of Mathematics, EMS Press
- J.D.M. Wright (2001) [1994], "AW* algebra", Encyclopedia of Mathematics, EMS Press