Absolutely s-pure and dual Kasch rings

First we show that a ring [Formula: see text] is right absolutely s-pure if and only if every finitely generated proper left ideal of [Formula: see text] has nonzero right annihilator. Then we define a ring [Formula: see text] to be a right US-ring if it is s-pure, as right [Formula: see text]-module, in every ring extension. Right absolutely s-pure rings are right US-rings and the converse is shown to be true when [Formula: see text] is commutative or right nonsingular. A counterexample is provided to prove that the reverse implication do not hold in general. We also prove that being right absolutely s-pure and being right dual Kasch are independent properties. Moreover, it is shown that these two properties are equivalent in the case when the ring is right and left quasi-duo. Finally, we give answers to two questions raised by Dobbs and Shapiro.