On Skew Polynomials over p.q.-Baer and p.p.-Modules

TL;DR

This paper investigates how skew polynomial extensions over modules preserve properties like p.q.-Baer and p.p.-modules under certain conditions, expanding understanding of module behavior in non-commutative polynomial contexts.

Contribution

It establishes new conditions under which p.q.-Baer and p.p.-modules are preserved in skew polynomial extensions, including the roles of conditions _1, _2, and _2 with _2.

Findings

Preservation of p.q.-Baer property under condition _2.

Equivalence of p.p.-module property between modules and their skew polynomial extensions.

Extension of results to semicommutative modules.

Abstract

Let $M_R$ be a module and $\sigma$ an endomorphism of $R$. Let $m\in M$ and $a\in R$, we say that $M_R$ satisfies the condition $\mathcal{C}_1$ (respectively, $\mathcal{C}_2$), if $ma=0$ implies $m\sigma(a)=0$ (respectively, $m\sigma(a)=0$ implies $ma=0$). We show that if $M_R$ is p.q.-Baer then so is $M[x;\sigma]_{R[x;\sigma]}$ whenever $M_R$ satisfies the condition $\mathcal{C}_2$, and the converse holds when $M_R$ satisfies the condition $\mathcal{C}_1$. Also, if $M_R$ satisfies $\mathcal{C}_2$ and $\sigma$-skew Armendariz, then $M_R$ is a p.p.-module if and only if $M[x;\sigma]_{R[x;\sigma]}$ is a p.p.-module if and only if $M[x,x^{-1};\sigma]_{R[x,x^{-1};\sigma]}$ ($\sigma\in Aut(R)$) is a p.p.-module. Many generalizations are obtained, and more results are found when $M_R$ is a semicommutative module.