Left-App rings of skew generalized power series

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 $a\in R$. Let $R$ be a ring, $(S,\leq)$ a strictly ordered monoid and $ω:S\longrightarrow {\rm End}(R)$ a monoid homomorphism. The skew generalized power series ring $[[R^{S,\leq},ω]]$ is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Malcev-Neumann Laurent series rings. We study the left APP-property of the skew generalized power series ring $[[R^{S,\leq},ω]]$. It is shown that if $(S,\leq)$ is a strictly totally ordered monoid, $ω:S\longrightarrow {\rm Aut}(R)$ a monoid homomorphism and $R$ a ring satisfying descending chain condition on right annihilators, then $[[R^{S,\leq},ω]]$ is left APP if and only if for any $S$-indexed subset $A$ of $R$, the ideal $l_{R}\big(\sum_{a\in A}\sum_{s\in S}Rω_{s}(a)\big)$ is right $s$-unital.

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