The elementary theory of ⟨α;×⟩, where α is an ordinal and × denotes ordinal multiplication, is decidable if and only if $\alpha <{\omega }^{\omega }$. Moreover if ${|}_r$ and ${|}_l$ respectively denote the right- and left-hand divisibility relation, we show that Th $⟨{\omega }^{{\omega }^{\xi }}{;|}_r⟩$ and Th $⟨{\omega }^{\xi }{;|}_l⟩$ are decidable for every ordinal ξ. Further related definability results are also presented.