# Rendiconti del Seminario Matematico della Università di Padova

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**Volume 133, 2015, pp. 91–102**

**DOI: 10.4171/RSMUP/133-4**

Published online: 2015-06-01

Pure injective and $\ast$-pure injective LCA groups

Peter Loth^{[1]}(1) Sacred Heart University, Fairfield, USA

A proper short exact sequence $0\to A\to B\to C\to 0$ in the category $\mathcal L$ of locally compact abelian (LCA) groups is called *$\ast$-pure* if the induced sequence $0\to A[n]\to B[n]\to C[n]\to 0$ is proper exact for all positive integers $n$. An LCA group is called *$\ast$-pure injective in $\mathcal L$* if it has the injective property relative to all $\ast$-pure sequences in $\mathcal L$. In this paper, we give a complete description of the $\ast$-pure injectives in $\mathcal L$. They coincide with the injectives in $\mathcal L$ and therefore with the pure injectives in $\mathcal L$. Dually, we determine the topologically pure projectives in $\mathcal L$.

*Keywords: *Locally compact abelian groups, pure injectives, $ast$-pure injectives, topologically pure projectives

Loth Peter: Pure injective and $\ast$-pure injective LCA groups. *Rend. Sem. Mat. Univ. Padova* 133 (2015), 91-102. doi: 10.4171/RSMUP/133-4