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I'm looking for a thorough introduction to group rings, specifically the simple case of group rings over the integers where the group is abelian and finitely generated. I realise that these are quotients of polynomial rings over the integers in finitely many variables, but I'm interested also in the perspective from group rings. Specifically I'd like to know, given such a group, methods to determine whether the group ring is connected, reduced and/or a domain, what its unit group is, etc.

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You may find useful these books:

  • Group Ring Groups by Eric Jespers and Ángel del Río

  • The Algebraic Structure of Group Rings by Donald Passman

  • Topics in group rings by Sudarshan Sehgal

  • An Introduction to Group Rings - Algebras & Applications by Sudarshan Sehgal and Cesar Polcino Milies

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    The first book was rather difficult to find, but each of these books is precisely what I was looking for. – 2017-02-01
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You can read "Groups, Rings and Galois Theory" by Victor. P. Snaith. It provides a good introduction on the topic. Also, "Groups, Rings and Modules" by Auslander and Buchsbaum is quite good. Hope it helps.

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    The table of contents of the first book doesn't mention group rings. From a glance I'd dare say I'm familiar with all (or most) of the material covered in it. The second book looks more promising. It's available at my library, I'll look it up later this week. – 2017-01-30
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    @Servaes Happy it helped. – 2017-01-30