On TFU extensions in LCA groups

Abstract

Let $\ell$ be the category of all locally compact abelian (LCA) groups. Let $G\in \ell$ and $H\subseteq G$. The first Ulm subgroup of $G$ is denoted by $G^{(1)}$ and the closure of $H$ by $\overline{H}$. A proper short exact sequence $0\to A\stackrel{\varphi }{\to }B\stackrel{\psi }{\to }C\to 0$ in $\ell$ is said to be a $TFU$ extension if $0\to \overline{A^{(1)}}\stackrel{\overline{\varphi }}{\to }\overline{B^{(1)}}\stackrel{\overline{\psi }}{\to }\overline{C^{(1)}}\to 0$ is a proper short exact sequence where $\overline{\varphi }=\varphi {\mid }_{\overline{A^{(1)}}}$ and $\overline{\psi }=\psi {\mid }_{\overline{B^{(1)}}}$. We introduce some results on $TFU$ extensions. Also, we establish conditions under which the $TFU$ extensions split.