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 [Formula: see text] be an integral domain with the field of fractions [Formula: see text] and let [Formula: see text] be an [Formula: see text]-algebra which is torsion-free as an [Formula: see text]-module. We show that if [Formula: see text] satisfies the three conditions: [Formula: see text] is generated by idempotents over [Formula: see text]; [Formula: see text] is countably infinite dimensional over [Formula: see text]; [Formula: see text] has [Formula: see text] primitive idempotents for a nonnegative integer [Formula: see text], then [Formula: see text] is uniquely determined up to [Formula: see text]-algebra isomorphism. We also consider the case where [Formula: see text] has countably many primitive idempotents.