[Paper Review] $\sigma$-PBW Extensions of Skew Armendariz Rings

Armando Reyes, Héctor Suárez|arXiv (Cornell University)|Nov 4, 2016

Rings, Modules, and Algebras16 references3 citations

TL;DR

This paper introduces and studies $σ$-PBW extensions of skew Armendariz rings, generalizing classical Armendariz properties to non-commutative algebras. It establishes that Baer, quasi-Baer, p.p., and p.q.-Baer properties lift from the base ring to the extension under skew-Armendariz conditions, extending results from Ore extensions and skew PBW extensions to a broader class of non-commutative rings with PBW bases.

ABSTRACT

The aim of this paper is to investigate a general notion of $\\sigma$-PBW extensions over Armendariz rings. As an application, the properties of being Baer, quasi-Baer, p.p. and p.q.-Baer are established for these extensions. We generalize several results in the literature for Ore extensions of injective type and skew PBW extensions.

Motivation & Objective

• To generalize the notion of Armendariz rings to $σ$-PBW extensions, which are non-commutative algebras with PBW bases.
• To extend known results on Baer, quasi-Baer, p.p., and p.q.-Baer properties from Ore extensions and skew PBW extensions to a broader class of non-commutative rings.
• To establish conditions under which these homological and ring-theoretic properties are preserved in $σ$-PBW extensions.
• To introduce and analyze two new notions: skew-Armendariz and weak skew-Armendariz rings in the context of $σ$-PBW extensions.

Proposed method

• Define $σ$-PBW extensions as rings $A$ containing a base ring $R$ with specific endomorphisms $\sigma_i$ and derivations $\delta_i$ acting on variables $X_i$.
• Introduce two new ring-theoretic conditions: skew-Armendariz (Definition 4.1) and weak skew-Armendariz (Definition 4.2), generalizing classical Armendariz properties.
• Use the notion of $(\Sigma,\Delta)$-ideals and $(\Sigma,\Delta)$-quasi-baer rings to analyze ideal structure and Baer-like properties in extensions.
• Prove that if $R$ is skew-Armendariz, then $R$ is Abelian and $\Sigma$-rigid, which enables lifting of Baer and p.p. properties to $A$.
• Apply results from previous works on skew PBW extensions (e.g., [46], [40], [49]) and adapt techniques from Ore extension theory ([39]) to $σ$-PBW settings.
• Use idempotent decomposition and annihilator conditions to characterize skew-Armendariz behavior in terms of corner rings $eR$ and $(1-e)R$.

Experimental results

Research questions

• RQ1 Under what conditions does the skew-Armendariz property of a base ring $R$ ensure that its $σ$-PBW extension $A$ inherits the Baer property?
• RQ2 How do the p.p.-ring and p.q.-Baer properties transfer from $R$ to $A$ when $R$ is skew-Armendariz?
• RQ3 Can the notion of Armendariz be generalized meaningfully to $σ$-PBW extensions beyond classical Ore extensions?
• RQ4 What is the relationship between the skew-Armendariz condition and the $IFP$ (insertion of factors property) in $σ$-PBW extensions?
• RQ5 To what extent are the families of skew-Armendariz and weak skew-Armendariz rings strictly larger than previously studied classes?

Key findings

• If $R$ is a skew-Armendariz ring, then $R$ is Abelian and $\Sigma$-rigid, which implies that $R$ is reduced and has trivial prime radical.
• The Baer property lifts from $R$ to $A$ if $R$ is skew-Armendariz: $R$ is Baer if and only if $A$ is Baer.
• The p.p.-ring property lifts from $R$ to $A$ under the same skew-Armendariz condition: $R$ is p.p. if and only if $A$ is p.p.
• The quasi-Baer and p.q.-Baer properties are preserved in $A$ if $R$ is skew-Armendariz and has the $IFP$ (insertion of factors property).
• The family of skew-Armendariz rings strictly contains the families studied in [40] and [49], showing a proper generalization.
• For an Abelian ring $R$, $R$ is (weak) skew-Armendariz if and only if for every idempotent $e$ with $\sigma_i(e)=e$ and $\delta_i(e)=0$, the corner rings $eR$ and $(1-e)R$ are (weak) skew-Armendariz.