If is a positive integer and is a prime, then a valuated -socle is said to be -summable if it is isometric to a valuated direct sum of countable valuated groups. The functions from to the cardinals that can appear as the Ulm function of an -summablevaluated -socle are characterized, as are the -summable valuated -socles that can appear as the -socle of some primary abeliangroup. The second statement generalizes a classical result of Honda from . Assuming a particular consequence of the generalized continuum hypothesis, a complete description is given of the -summable groups that are uniquely determined by their Ulm functions.
Cite this article
Patrick W. Keef, Realization Theorems for Valuated -Socles. Rend. Sem. Mat. Univ. Padova 126 (2011), pp. 151–173DOI 10.4171/RSMUP/126-9