Residual and fixed modules

Abstract

The article presents some sufficient conditions for the commutativity of transvections with elements of linear groups over division ring in the language of residual and fixed submodules. The residual and fixed submodules of the element $\sigma $ of the linear group are defined as the image and nucleus of the element $\sigma -1$ and are denoted by $R(\sigma)$ and $P(\sigma)$ respectively. It is proved that transvection ${\sigma }_1$ over an arbitrary body commutes with an element ${\sigma }_2$ for which $\mathop{\rm dim}R({\sigma }_2)=\mathop{\rm dim}R({\sigma }_2)\cap P({\sigma }_2)+l$, $l\le 1$, if and only if the inclusion system $R({\sigma }_1)\subseteq P({\sigma }_2)$, $R({\sigma }_2)\subseteq P({\sigma }_1)$. It is shown that for $l>1$ this statement is not always true.