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Let G be an infinite abelian p-group and let K be a field of the first kind with respect to p of characteristic different from p such that $s_p\left(K\right)=\mathbb{N}$ or $s_p\left(K\right)=\mathbb{N}\cup 0$. The main result of the paper is the computation of the Ulm-Kaplansky functions of the factor group S(KG)/G of the normalized Sylow p-subgroup S(KG) in the group ring KG modulo G. We also characterize the basic subgroups of S(KG)/G by proving that they are isomorphic to S(KB)/B, where B is a basic subgroup of G.