Baer ring

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.) An idempotent element of a ring is an element e which has the property that e2 = 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. 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. For unital rings, replacing all occurrences of 'left' with 'right' yields an equivalent definition, that is to say, the definition is left-right symmetric. 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 Rop, 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.