Right ideal decompositions of G⁎

Let G be an infinite discrete group, let βG be the Stone–Čech compactification of G  , and let G⁎=βG∖GG⁎=βG∖G. We show that if G   can be embedded algebraically in a compact zero dimensional second countable group (in particular, if G=ZG=Z), then there are a decomposition DD of G⁎G⁎ into right ideals of βG and a closed subsemigroup T   of G⁎G⁎ containing all the idempotents such that DT={R∩T:R∈D}DT={R∩T:R∈D} is a decomposition of T   into closed right ideals and T/DTT/DT is homeomorphic to ω⁎ω⁎.