• An
idempotent element of a ring is an element
e which has the property that
e2 =
e. • The
left annihilator of a set X \subseteq R is \{r\in R\mid rX=\{0\}\} • 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 *:R\rightarrow R. 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 (). • 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. ==Examples==