# Realization Theorems for Valuated $p^n$-Socles

### Patrick W. Keef

Whitman College, Walla Walla, USA

## Abstract

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.

## Cite this article

Patrick W. Keef, Realization Theorems for Valuated $p^n$-Socles. Rend. Sem. Mat. Univ. Padova 126 (2011), pp. 151–173

DOI 10.4171/RSMUP/126-9