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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== *An idempotent element of a ring is an element ''e'' which has the property that ''e''2 = ''e''. *The left annihilator of a set is *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.〔Rickart rings are named after who studied a similar property in operator algebras. This "principal implies projective" condition is the reason Rickart rings are sometimes called PP-rings. 〕 *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''.〔This condition was studied by .〕 For unital rings, replacing all occurrences of 'left' with 'right' yields an equivalent definition, that is to say, the definition is left-right symmetric.〔T.Y. Lam (1999), "Lectures on Modules and Rings" ISBN 0-387-98428-3 pp.260〕 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 ''R''op, 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 , is a C * algebra that is also a Baer *-ring. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Baer ring」の詳細全文を読む スポンサード リンク
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