A Baer ring is a ring R in which, for every subset X of R, the right annihilator of X, denoted annr(X), is generated as a right ideal by an idempotent element. In other words, for every subset X ⊆ R, there exists an idempotent e ∈ R such that annr(X) = eR.
Equivalently, a Baer ring is a ring in which the left annihilator of every subset is generated as a left ideal by an idempotent. This definition is left-right symmetric.
Important classes of rings are Baer rings:
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von Neumann algebras: Every von Neumann algebra is a Baer ring. This was the original motivation for the definition of Baer rings.
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AW-algebras:* Every AW*-algebra is a Baer ring.
Some related concepts:
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Rickart ring: A ring R is a Rickart ring if, for every element x ∈ R, the right annihilator of x is generated by an idempotent. Baer rings generalize Rickart rings. In other words, a Rickart ring is a ring where the right annihilator of a singleton set is generated by an idempotent.
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*Baer -ring: A Baer *-ring is a ring with an involution * that is also a Baer ring.
Baer rings are important in functional analysis and operator algebras, providing an algebraic framework for studying completeness properties and projection lattices of operator algebras.