On absolute 1-purity of acts

In this paper we investigate the actions of a monoid of the form S = G ˙[I, where G is a group and I is an ideal of S , on sets. So, naturally, every S -act can be considered as an I1-act. The central question here is that what is the relation between absolute 1-purity of I1-acts and absolute 1-purity of S -acts? Here we respond to this question and show that we need some more hypothesis.