Left-App rings of skew generalized power series

TL;DR

This paper investigates the conditions under which skew generalized power series rings possess the left APP property, extending the understanding of their structure and relation to other ring classes.

Contribution

It characterizes the left APP property of skew generalized power series rings in terms of right s-unitality of certain annihilator ideals under specific conditions.

Findings

Left APP property characterized by right s-unitality of annihilator ideals.

Applicable to various classical rings like polynomial and Laurent rings.

Provides necessary and sufficient conditions for left APP in skew power series rings.

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