An Extension of Nice Bases on Ulm Subgroups of Primary Abelian Groups

Suppose A is an Abelian p-group and is an ordinal such that A/p^α A is a direct sum of countable groups. It is shown that A has a nice basis if, and only if, p^α A has a nice basis. This strengthens an earlier result of ours in Bull. Allah. Math. Soc. (2011) proved when α< ω².