Isomorphism classes of commutative algebras generated by idempotents

We study commutative algebras generated by idempotents with particular emphasis on the number of primitive idempotents. Let R be an integral domain with the field of fractions K and let A be an R-algebra which is torsion-free as an R-module. We show that if A satisfies the three conditions: A is generated by idempotents over R; K circle times RA is countably infinite dimensional over K; A has n primitive idempotents for a nonnegative integer n, then A is uniquely determined up to R-algebra isomorphism. We also consider the case where A has countably many primitive idempotents.