Realization Theorems for Valuated $p^n$-Socles

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 $n$-summable valuated $p^n$-socle are characterized, as are the $n$-summable valuated $p^n$-socles that can appear as the $p^n$-socle of some primary abelian group. 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.