A note on $\sigma$-reversibility and $\sigma$-symmetry of skew power series rings

TL;DR

This paper investigates how properties like $\sigma$-symmetry and $\sigma$-reversibility transfer from a ring $R$ to its skew power series ring $R[[x;\sigma]]$, and explores related ring properties under these conditions.

Contribution

It provides new insights into the transfer of symmetry and reversibility properties to skew power series rings and generalizes previous results in the literature.

Findings

Transfer of $\sigma$-symmetry and $\sigma$-reversibility to skew power series rings.

Relationship between Baerness, quasi-Baerness, and p.p.-properties of $R$ and $R[[x;\sigma]]$.

Generalization of previous results on ring properties under $\sigma$-reversibility.

Abstract

Let $R$ be a ring and $\sigma$ an endomorphism of $R$. In this note, we study the transfert of the symmetry ($\sigma$-symmetry) and reversibility ($\sigma$-reversibility) from $R$ to its skew power series ring $R[[x;\sigma]]$. Moreover, we study on the relationship between the Baerness, quasi-Baerness and p.p.-property of a ring $R$ and these of the skew power series ring $R[[x;\sigma]]$ in case $R$ is right $\sigma$-reversible. As a consequence we obtain a generalization of \cite{hong/2000}.