On the number of generators of the module of derivations and multiplicity of certain rings

The present paper studies the module of derivations of certain rings and the multiplicity of certain rational surface singularities. We also consider some relation between these two for 2-dimensional quotient singularities. We conjecture a relation between the multiplicity of a rational surface singularity and the order of the divisor class group of the singularity and verify the same for several cases.