On commutative von Neumann regular rings and a question of Handelman Original Research Article

The main purpose of this note is to answer a question of Handelman on right self-injective, directly finite, von Neumann regular rings. He asked if such a ring R is determined up to isomorphism by the Boolean space Spec B (R), where B( R) is the boolean algebra of central idempotents of R, and the factor rings R/Ix, for x ε Spec B(R) and Ix the unique maximal ideal of R containing x. The answer is “no”, even for commutative self-injective rings. An example is constructed using the complete ring of quotients of two rings of sections of simple Pierce sheaves over an extremally disconnected space.