# Looking for a concise treatise on the $\mathbb Z^n$ as a $\mathbb Z$-module

I am working on something which turned out to be isomorphic to the $\mathbb Z$-module $\mathbb Z^n$ and want to efficiently learn about its properties without needing to deal too much with more general modules or rings. I am therefore looking for a concise book, book chapter, review paper or similar on the $\mathbb Z^n$ and its module structure and features such as bases, linear maps etc.

I realise that this might be an utopistic wish, so I also welcome something that has a broader scope, e.g., $\mathbb R^n$ with $\mathbb R$ being a ring sharing many properties with $\mathbb Z$.

My prerequisites are:

• I consider myself to be knowledgable on linear algebra including more abstract topics.
• I have little knowledge of groups, rings, etc.
• English and German are fine as a language.
• "module"...over what ring? – DonAntonio Mar 3 '14 at 12:35
• Related: en.wikipedia.org/wiki/Free_abelian_group All $\mathbb{Z}$ modules are abelian groups. – Tim Seguine Mar 3 '14 at 12:37
• @DonAntonio: $\mathbb{Z}$ – Wrzlprmft Mar 3 '14 at 12:37
• @Wrzlprmft, then it is exactly the same as an abelian group: $\;\Bbb Z^n\;$ is the free abelian group of rank $\;n\;$ and any abelian group with up to $\;n\;$ generators is a homomorphic image of it. – DonAntonio Mar 3 '14 at 12:39
• Dear @Wrzlprmft : To be clear, DonAntonio and Tim are saying that $\Bbb Z$ modules are exactly abelian groups and that $\Bbb Z^n$ is a particular finitely generated abelian group. It wouldn't be right to say that things of the form $\Bbb Z^n$ are "equivalent" to finitely generated abelian groups. – rschwieb Mar 5 '14 at 14:40

A good choice for a type of ring "sharing many properties with $\Bbb Z$" would be any principal ideal domain, and there is a simple classification theorem for finitely generated modules over such rings.
You've picked an even narrower subclass of finitely generated $\Bbb Z$ modules: that of the finitely generated free $\Bbb Z$-modules. Quite nicely, the submodules of such modules are again free modules. Concretely, the submodules of $\Bbb Z^n$ look like $\Bbb Z^k$ where $k\leq n$.
Free modules are about as nice as you can get. They have "bases" analogous to those of bases of vector spaces, and the $\Bbb Z$ linear transformations of $\Bbb Z^n\to\Bbb Z^m$ is again described by the matrices $M_{n\times m}(\Bbb Z)$.
Whatever your application is, it might be worthwhile to extend your scope to all finitely generated $\Bbb Z$-modules, a.k.a. finitely generated abelian groups. This is your best bet for what to look up in references. Most basic algebra books include the structure theorem of finitely generated abelian groups, which is just a special case of the one I mentioned for PIDs.