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 $0\to A\stackrelφ{\to} B\stackrelψ{\to} C\to 0$ in $\ell$ is said to be a $TFU$ extension if $0\to \overline{A^{(1)}}\stackrel{\overlineφ}{\to} \overline{B^{(1)}}\stackrel{\overlineψ}{\to} \overline{C^{(1)}}\to 0$ is a proper short exact sequence where $\overlineφ=φ\mid_{\overline{A^{(1)}}}$ and $\overlineψ=ψ\mid_{\overline{B^{(1)}}}$. We introduce some results on $TFU$ extensions. Also, we establish conditions under which the $TFU$ extensions split.

Theorems & Definitions (33)

• Definition 2.1
• Remark 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Corollary 2.4
• proof
• Lemma 2.5
• proof