INSERTION-OF-FACTORS-PROPERTY SKEWED BY RING ENDOMORPHISMS

Abstract

In this paper, we investigate the Insertion-of-Factors-Property (simply IFP), (quasi-)Baer property, and Armendariz property on skew power series (polynomial) rings and introduce the concept of (strongly) $\sigma$-skew IFP and extend many of related basic results to the wider classes. When a ring $R$ has $\sigma$-skew IFP and $\sigma$ is a monomorphism of $R$ we prove that $R$ is Baer if and only if $R$ is quasi-Baer if and only if $R[[x;\sigma]]$ ($R[x;\sigma]$) is Baer if and only if $R[[x;\sigma]]$ ($R[x;\sigma]$) is quasi-Baer. We also prove that if $R$ is a skew power-serieswise $\sigma$-Armendariz ring then $R$ has strongly $\sigma$-skew IFP and $R[[x;\sigma]]$ has IFP. Several known results follow as consequences of our results. In particular, we provide a $\sigma$-skew power-serieswise Armendariz ring but does not have IFP.