The (first) Ulm subgroup of an abelian group A, denoted U(A) or A1, is pωA = ∩n pnA, where ω is the smallest infinite ordinal.
There is a complement to this theorem, first stated by Leo Zippin (1935) and proved in Kurosh (1960), which addresses the existence of an abelian p-group with given Ulm factors.
Ulm's original proof was based on an extension of the theory of elementary divisors to infinite matrices.
George Mackey and Irving Kaplansky generalized Ulm's theorem to certain modules over a complete discrete valuation ring.
Their simplified proof of Ulm's theorem served as a model for many further generalizations to other classes of abelian groups and modules.