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== *An idempotent element of a ring is an element ''e'' which has the property that ''e''<sup>2</sup> = ''e''. *The '''left annihilator''' of a set <math>X \subseteq R</math> is <math>\{r\in R\mid rX=\{0\}\}</math> *A '''(left) Rickart ring''' is a ring satisfying any of the following conditions: # the left annihilator of any single element of ''R'' is generated (as a left ideal) by an idempotent element. # (For unital rings) the left annihilator of any element is a direct summand of ''R''. # All principal left ideals (ideals of the form ''Rx'') are projective ''R'' modules.<ref>Rickart rings are named after {{harvtxt|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. {{harv|Lam|1999}}</ref> *A '''Baer ring''' has the following definitions: # The left annihilator of any subset of ''R'' is generated (as a left ideal) by an idempotent element. # (For unital rings) The left annihilator of any subset of ''R'' is a direct summand of ''R''.<ref>This condition was studied by {{harvs|txt|authorlink=Reinhold Baer|first=Reinhold |last=Baer|year=1952}}.</ref> For unital rings, replacing all occurrences of 'left' with 'right' yields an equivalent definition, that is to say, the definition is left-right symmetric.<ref>T.Y. Lam (1999), "Lectures on Modules and Rings" {{ISBN|0-387-98428-3}} pp.260</ref>
In operator theory, the definitions are strengthened slightly by requiring the ring ''R'' to have an involution <math>*:R\rightarrow R</math>. Since this makes ''R'' isomorphic to its opposite ring ''R''<sup>op</sup>, the definition of Rickart *-ring is left-right symmetric. * A '''projection''' in a *-ring is an idempotent ''p'' that is self-adjoint ({{nowrap|1=''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 {{harvtxt|Kaplansky|1951}}, is a C*-algebra that is also a Baer *-ring.
==Examples==
*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 <math>\{0\}</math> except for the annihilator of 0, which is ''R'', and both <math>\{0\}</math> 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.
==Properties==
The projections in a Rickart *-ring form a lattice, which is complete if the ring is a Baer *-ring.
== See also == * Baer *-semigroup
==Notes== {{Reflist}}
==References==
* {{Citation | last1=Baer | first1=Reinhold | author1-link=Reinhold Baer | title=Linear algebra and projective geometry| publisher=Academic Press | location=Boston, MA | mr=0052795 | year=1952}}; reprinted, Dover Publications, 2005, {{isbn|978-0-486-44565-6}} * {{Citation | last1=Berberian | first1=Sterling K. | title=Baer *-rings | url=https://books.google.com/books?isbn=354005751X | publisher=Springer-Verlag | location=Berlin, New York | series=Die Grundlehren der mathematischen Wissenschaften | isbn=978-3-540-05751-2 | mr=0429975 | year=1972 | volume=195}} * {{Citation | last1=Kaplansky | first1=Irving | author1-link=Irving Kaplansky | title=Projections in Banach algebras | jstor=1969540 | mr=0042067 | year=1951 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=53 | pages=235–249 | issue=2 | doi=10.2307/1969540}} * {{citation|first=I.|last= Kaplansky|title=Rings of Operators|publisher=W. A. Benjamin, Inc.|place= New York|year= 1968|url=https://books.google.com/books?id=hRaoAAAAIAAJ}} *{{Citation |last1=Lam |first1=Tsit-Yuen |author-link=Tsit Yuen Lam |title=Lectures on modules and rings |publisher=Springer-Verlag |location=Berlin, New York |series=Graduate Texts in Mathematics No. 189 |isbn=978-0-387-98428-5 |mr=1653294 |year=1999}} * {{citation|last=Rickart|first= C. E.|authorlink=Charles Earl Rickart|title=Banach algebras with an adjoint operation|jstor=1969091|journal=Annals of Mathematics |series=Second Series|volume=47|year=1946|pages=528–550|mr=0017474|issue=3|doi=10.2307/1969091}} *{{springer|id=R/r080830|title=Regular ring (in the sense of von Neumann)|author=L.A. Skornyakov}} *{{springer|id=R/r081840|title=Rickart ring |author=L.A. Skornyakov}} *{{springer|id=A/a120310|title=AW* algebra|author=J.D.M. Wright}}
Category:Von Neumann algebras Category:Ring theory