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| In [[abstract algebra]] and [[functional analysis]], '''Baer rings''', '''Baer *-rings''', '''Rickart rings''', '''Rickart *-rings''', and '''[[AW*-algebra]]s''' are various attempts to give an algebraic analogue of [[von Neumann algebra]]s, using axioms about [[annihilator]]s of various sets.
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| 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.
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| In the literature, left Rickart rings have also been termed left '''PP-rings'''. ("Principal implies projective": See definitions below.)
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| ==Definitions==
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| *An [[idempotent element]] of a ring is an element ''e'' which has the property that ''e''<sup>2</sup> = ''e''.
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| *The '''left [[annihilator]]''' of a set <math>X \subseteq R</math> is <math>\{r\in R\mid rX=\{0\}\}</math>
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| *A '''(left) Rickart ring''' is a ring satisfying any of the following conditions:
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| # the left annihilator of any single element of ''R'' is generated (as a left ideal) by an idempotent element.
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| # (For unital rings) the left annihilator of any element is a direct summand of ''R''.
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| # All principal left ideals (ideals of the form ''Rx'') are [[projective module|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>
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| *A '''Baer ring''' has the following definitions:
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| # The left annihilator of any subset of ''R'' is generated (as a left ideal) by an idempotent element.
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| # (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>
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| In operator theory, the definitions are strengthened slightly by requiring the ring ''R'' to have an [[ring with involution|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.
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| * A '''projection''' in a [[*-ring]] is an idempotent ''p'' that is self adjoint (''p''*=''p'').
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| *A '''Rickart *-ring''' is a *-ring such that left annihilator of any element is generated (as a left ideal) by a projection.
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| *A '''Baer *-ring''' is a *-ring such that left annihilator of any subset is generated (as a left ideal) by a projection.
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| *An '''AW* algebra''', introduced by {{harvtxt|Kaplansky|1951}}, is a [[C* algebra]] that is also a Baer *-ring.
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| ==Examples==
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| *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 ring]]s, which are left and right semihereditary. If a von Neumann regular ring ''R'' is also right or left [[injective module#self injective rings|self injective]], then ''R'' is Baer.
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| *Any [[semisimple ring]] is Baer, since ''all'' left and right ideals are summands in ''R'', including the annihilators.
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| *Any [[domain (ring theory)|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''.
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| *The ring of [[bounded linear operator]]s on a [[Hilbert space]] are a Baer ring and is also a Baer *-ring with the involution * given by the adjoint.
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| *von Neumann algebras are examples of all the different sorts of ring above.
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| ==Properties==
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| The projections in a Rickart *-ring form a [[lattice (order)|lattice]], which is [[complete lattice|complete]] if the ring is a Baer *-ring.
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{Citation | last1=Baer | first1=Reinhold | author1-link=Reinhold Baer | title=Linear algebra and projective geometry | url=http://books.google.com/books?isbn=012072250X | publisher=[[Academic Press]] | location=Boston, MA | isbn=978-0-486-44565-6 | mr=0052795 | year=1952}}
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| *{{Citation | last1=Berberian | first1=Sterling K. | title=Baer *-rings | url=http://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}}
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| *{{Citation | last1=Kaplansky | first1=Irving | author1-link=Irving Kaplansky | title=Projections in Banach algebras | jstor=1969540 | mr=0042067 | year=1951 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=53 | pages=235–249 | issue=2 | doi=10.2307/1969540}}
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| *{{citation|first=I.|last= Kaplansky|title=Rings of Operators|publisher=W. A. Benjamin, Inc.|place= New York|year= 1968|url=http://books.google.com/books?id=hRaoAAAAIAAJ}}
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| *{{citation|last=Rickart|first= C. E.|title=Banach algebras with an adjoint operation|jstor=1969091|journal=Annals of Mathematics. Second Series|volume=47|year=1946|pages=528–550|mr=0017474|issue=3|doi=10.2307/1969091}}
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| *{{springer|id=R/r080830|title=Regular ring (in the sense of von Neumann)|author=L.A. Skornyakov}}
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| *{{springer|id=R/r081840|title=Rickart ring |author=L.A. Skornyakov}}
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| *{{springer|id=A/a120310|title=AW* algebra|author=J.D.M. Wright}}
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| <references/>
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| [[Category:Von Neumann algebras]]
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| [[Category:Ring theory]]
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