Denote a (countable) direct product of Euclidean domains by \(\Pi\) (viewed both as a ring as well as its underlying additive group), the additive subgroup of \(\Pi\), \({\mathbb{Z}}\)-generated by all the characteristic elements of \(\Pi\) by B, and, by \(\omega\), a subring of \(\Pi\) which contains B. The authors study the \(\omega\)-pure submodules of \(\omega\), direct sums of direct summands of \(\omega\) and direct summands of \(\Pi\) in particular. They prove that the study of pure \(\omega\)-submodules of \(\omega\) may be reduced to that of pure B-submodules of B.