Commutative/noncommutative algebra? I know basic knowledge of undergraduate algebra till galois theory of finite extensions. I want to learn number theory, but also like algebra. This semester I have to choose to read either commutative algebra or noncommutative algebra. Can somebody tell me briefly the nature of both and what they lead? i want to know what they deal with and also in what way they are relevant in number theory. Thanks!
 A: I'll also point out that commutative algebra is at least half the foundation of modern algebraic geometry. The objects of algebraic geometry are built by sticking together a collection of geometric structures associated to a commutative ring-so to study them, one must know plenty of commutative algebra. And if you're interested in going very far into number theory, you'll likely want to learn some algebraic geometry eventually.
A: If you're interested in number theory, the commutative algebra will be much more helpful to you in the immediate future. Knowing some amount of noncommutative algebra (i.e. Wedderburn theory) will be needed later in your mathematical life, but right now you need to know things about Dedekind domains. This entails knowing what the words Krull dimension, Noetherian, and integrally closed means. These are all commutative algebra terms.
Oh, to add about "what they're about", let me say the following. Commutative algebra is, well, the study of commutative rings. It's relation to (algebraic) number theory, is what I mentioned above. The key objects of study in basic number theory are number rings $\mathcal{O}_K$. The mere definition of these objects requires the commutative algebra notion of "integral closure". The beginning of the whole subject of ANT is the realization that number rings are special types of rings called "Dedekind domains". 
These have the nice definition of being the domains with unique factorization of ideals into prime ideals. While that definition is nice, it's cumbersome to prove things with. Thus, one often adopts the equivalent definition that a Dedekind domain is a domain which is dimension 1, Noetherian, and integrally closed. Once again, the mere definitions of these words are commutative algebraic in nature, and their study, and the subsequent application of these ideas to number rings requires some non-trivial amount of knowledge of commutative algebra. 
Also, commutative algebra is just a very functional part of modern mathematics, and is important to know just to be able to speak intelligibly about a lot of topics.

Noncommutative algebra, at least in its standard meaning, is the study of non-commutative rings and the resulting theory. This is slightly more obscure, and comes up in number theory much later. 
The main application in semi-basic number theory that I can think of is the study of (relative) Brauer groups of a field $K$. In particular, there one studies central simple algebras over $K$ up to something called Morita equivalence. Once again, the very nature of these words is noncommutative algebraic, and one of the pivotal theorems in their basic study is that of Weddurburn-Artin theorem which classifies semisimple algebras (as well as some of the surrounding theory [e.g. Noether-Skolem]). 
But, as I said, this doesn't come up until much later in one's basic number theory career (it's really seen in something called "class field theory").
Also, while noncommutative algebra is a very important took in mathematics, especially more advanced mathematics, it is less fundamental in early mathematics. This may be, perhaps, because of my particular interests, but really does seem to be a function of what is in vogue right now.

So, to summarize the above. Commutative algebra will not only be more useful to you in number theory sooner, but will transfer more readily, and usefully, to other areas of mathematics you are likely to learn in the near future.
