Socle chains of abelian regular semiartinian rings

Let R be an abelian regular and semiartinian ring with socle chain (Sα∣α≤σ). If λα denotes the rank of the semisimple module Sα+1/Sα, for every α<σ, then the dimension sequence (λα∣α<σ) is an invariant for R. By applying classical results of combinatorial set theory we prove necessary conditions satisfied by this invariant. On the other hand, we present constructions of commutative regular semiartinian rings with given ranks of slices of socle chain. In some particular cases we prove a necessary and sufficient condition under which there exists an abelian regular semiartinian ring with given ranks of slices.