%0 Journal Article
%T On T-Extensions of Abelian Groups
%J Khayyam Journal of Mathematics
%I Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with the Center of Excellence in Analysis on Algebraic Structures)
%Z 2423-4788
%A Alijani, Aliakbar
%A Sahleh, Hossein
%D 2019
%\ 01/01/2019
%V 5
%N 1
%P 60-68
%! On T-Extensions of Abelian Groups
%K T-extensions
%K extensions
%K pure extensions
%K locally compact abelian groups
%R 10.22034/kjm.2018.74220
%X Let $\Re$ be the category of all discrete abelian groups, and let $\cal{L}$ be the category of all locally compact abelian (LCA) groups. For a group $G\in \cal{L}$, the maximal torsion subgroup of $G$ is denoted by $tG$. A short exact sequence $0\to A\stackrel{\phi}{\to} B\stackrel{\psi}{\to}C\to 0$ in $\Re$ is said to be a t-extension if $0\to tA\stackrel{\phi}{\to} tB\stackrel{\psi}{\to}tC\to 0$ is a short exact sequence. We show that the set of all t-extensions of $A$ by $C$ is a subgroup of $Ext(C,A)$, which contains $Pext(C,A)$ for discrete abelian groups $A$ and $C$. We establish conditions under which the t-extensions split and determine those groups in $\Re$ which are t-injective or t-projective in $\Re$. Finally we determine the compact groups $G$ in $\cal{L}$ such that every pure extension of $G$ by a compact connected group $C\in \cal{L}$ splits.
%U https://www.kjm-math.org/article_74220_a9f5d5879c28658ed606d177fed55aa0.pdf