Two generalizations of T-nilpotence

A commutative ring R is called T-idempotent if, for every sequence r1,r2,⋯ ∈ R, there exists a positive integer n and an idempotent e ∈ R such that r1 ⋯rn = e. A commutative ring R is called T-stable if, for every sequence r1,r2,⋯ R, there exists a positive integer n such that r1⋯rn = r1⋯rn+1 = ⋯. We show that a commutative ring is T-idempotent if and only if it is the direct product of a T-nilpotent ring and a Boolean ring. We prove that a commutative ring is T-stable if and only if it is the direct product of a T-nilpotent ring and a Boolean ring satisfying the descending chain condition for idempotents. We describe all commutative T-idempotent and T-stable semigroup rings. © Springer-Verlag 1998.