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European Mathematical Society Publishing House
2016-09-19 17:05:48
Rendiconti del Seminario Matematico della Università di Padova
Rend. Sem. Mat. Univ. Padova
RSMUP
0041-8994
2240-2926
General
10.4171/RSMUP
http://www.ems-ph.org/doi/10.4171/RSMUP
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European Mathematical Society Publishing House
Zuerich, Switzerland
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126
2011
0
Realization Theorems for Valuated $p^n$-Socles
Patrick
Keef
Whitman College, WALLA WALLA, UNITED STATES
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.
General
151
173
10.4171/RSMUP/126-9
http://www.ems-ph.org/doi/10.4171/RSMUP/126-9