# Rendiconti del Seminario Matematico della Università di Padova

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**Volume 126, 2011, pp. 151–173**

**DOI: 10.4171/RSMUP/126-9**

Realization Theorems for Valuated $p^n$-Socles

Patrick W. Keef^{[1]}(1) Whitman College, Walla Walla, USA

If $n$ is a positive integer and $p$ is a prime, then a valuated $p^n$-socle is said to be $n$-summable if it is isometric to a valuated direct sum of countable valuated groups. The functions from $\omega_1$to the cardinals that can appear as the Ulm function of an $n$-summablevaluated $p^n$-socle are characterized, as are the $n$-summable valuated $p^n$-socles that can appear as the $p^n$-socle of some primary abeliangroup. The second statement generalizes a classical result of Honda from [9]. Assuming a particular consequence of the generalized continuum hypothesis, a complete description is given of the $n$-summable groups that are uniquely determined by their Ulm functions.

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Keef Patrick: Realization Theorems for Valuated $p^n$-Socles. *Rend. Sem. Mat. Univ. Padova* 126 (2011), 151-173. doi: 10.4171/RSMUP/126-9