Left-App rings of skew generalized power series

Abstract

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, $(S,\le )$ a strictly ordered monoid and $\omega :S⟶\mathrm{End}(R)$ a monoid homomorphism. The skew generalized power series ring $[[R^{S,\le },\omega ]]$ 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 $[[R^{S,\le },\omega ]]$. It is shown that if $(S,\le )$ is a strictly totally ordered monoid, $\omega :S⟶\mathrm{Aut}(R)$ a monoid homomorphism and $R$ a ring satisfying descending chain condition on right annihilators, then $[[R^{S,\le },\omega ]]$ is left APP if and only if for any $S$-indexed subset $A$ of $R$, the ideal $l_R({\sum }_{a\in A}{\sum }_{s\in S}R{\omega }_s(a))$ is right $s$-unital.

10 pages