I'm looking for a modern, approachable text (preferably a website, textbook, or expository article, and preferably one easily available online or at a library) which can explain the concept of Ulm invariants and how they factor into the classification of (countable) abelian groups.
The theorem I'm trying to understand is the following, apparently due to Mackey and Kaplansky:
A countable periodic reduced abelian group is determined uniquely up to isomorphism by its Ulm invariants for all prime numbers p and countable ordinals α.
Google covered the definitions of periodic (all elements of finite order; with abelian this is equivalent to locally finite) and reduced (no divisible elements but zero), but the definition of Ulm invariant is still a bit dodgy. The proof itself is not summarized, so it's unclear to what extent the following questions are answered or unanswered:
What if the group is not reduced? Do we have a characterization of (countable) divisible periodic abelian groups?
(apparently the divisible part is always a direct summand, so it's enough to do each part separately)
What if the group is not periodic? Can we use (e.g.) the finitely generated abelian groups theorem to sort of separate out the $\mathbb Z$-pieces?
What if the group is not countable? Can we see it canonically as made up of countable pieces, and assemble them?
I should say that I'm a logician, not an algebraist, so I don't know the "standard sources" in this field. But I'm looking to learn.