Left APP-property of formal power series rings

summary:A ring $R$ is called a left APP-ring if the left annihilator $l_R(Ra)$ is right $s$-unital as an ideal of $R$ for any element $a\in R$. We consider left APP-property of the skew formal power series ring $R[[x; \alpha ]]$ where $\alpha $ is a ring automorphism of $R$. It is shown that if $R$ is a ring satisfying descending chain condition on right annihilators then $R[[x; \alpha ]]$ is left APP if and only if for any sequence $(b_0, b_1, \dots )$ of elements of $R$ the ideal $l_R$ $\big (\sum _{j=0}^{\infty }\sum _{k=0}^{\infty }R\alpha ^k(b_j)\big )$ is right $s$-unital. As an application we give a sufficient condition under which the ring $R[[x]]$ over a left APP-ring $R$ is left APP