Question is as it is stated in title.
I will be joining for PhD program in this July 2014.
I am interested in working in Algebra/Algebraic Geometry/Algebraic Number Theory.
I tried to learn algebra from Serge Lang's book (some two and half years back), but due to lack of background, I could not understand a bit of it, and I lost interest.
I always wanted to read it, but because I could not understand anything in it, and because most of my seniors keep saying "Lang is difficult," I lost interest and hope in reading that. I easily get irritated by seeing that book.
One of my friend gave me his copy of Abstract Algebra by Dummit and Foote. It was totally different from Lang, and I was comfortable reading that. Now I have done almost all exercises in three fourths of the book (with help of MSE).
The curriculum for coursework in the coming year is:
Review of field and Galois theory: solvable and radical extensions, Kummer theory, Galois cohomology and Hilbert's Theorem 90, Normal Basis theorem.
Infinite Galois extensions: Krull topology, projective limits, profinite groups, Fundamental Theorem of Galois theory for infinite extensions.
Review of integral ring extensions: integral Galois extensions, prime ideals in integral ring extensions, decomposition and inertia groups, ramification index and residue class degree, Frobenius map, Dedekind domains, unique factorisation of ideals.
Categories and functors: definitions and examples. Functors and natural transformations, equivalence of categories,. Products and coproducts, the hom functor, representable functors, universals and adjoints. Direct and inverse limits. Free objects.
Homological algebra: Additive and abelian categories, Complexes and homology, long exact sequences, homotopy, resolutions, derived functors, Ext, Tor, cohomology of groups, extensions of groups.
Valuations and completions: definitions, polynomials in complete fields (Hensel's Lemma, Krasner's Lemma), finite dimensional extensions of complete fields, local fields, discrete valuations rings.
Transcendental extensions: transcendence bases, separating transcendence bases, Luroth's theorem. Derivations.
Artinian and Noetherian modules, Krull-Schmidt theorem, completely reducible modules, projective modules, Wedderburn-Artin Theorem for simple rings.
Representations of finite groups: complete reducibility, characters, orthogonal relations, induced modules, Frobenius reciprocity, representations of the symmetric group.
The suggested book for this is S. Lang, Algebra, 3rd Ed., Addison Wesley, 1993.
I do not know if I have to choose some other book or convince myself (I do not know how) to be with Lang's book. I want to remind again that I have no motivation to.
Another thing that I heard is that it is better to use Lang as reference book than a textbook for a course.
I am in a confused state.
I really disliked Lang's book, at first. His depth is comprehensive, but he jumps into most topics at the maximum level of complication and detail, which is extremely jarring for someone just learning the material. On top of that, the topics are sometimes somewhat out of order, with examples from one chapter coming from material from a later chapter (though this can be said of Dummit and Foote as well). It can be very frustrating trying to learn anything from this book if you've never seen it before.
With this said, when trying to study the same topics for my course from other books, I found myself eventually coming back to Lang. Other sources proved to be good introductions to a subject, but few approached the depth that Lang eventually got to. I eventually came to really appreciate the vast array of examples Lang included, even though some of them were above my head at first. You could do what I did, too: read other books when first learning a topic, get a basic grasp of it, then transition to Lang to finish. However, it can be hard to keep terminology straight during these transitions, so the best approach may be to just work through Lang slowly, realizing that you won't be able to understand right away and that that's okay.
Anyway, if you do choose to replace Lang with something else, it shouldn't be Dummit and Foote. Dummit and Foote is a great book, and you'd do well to read it concurrently, but it's a level below Lang. Many of the topics you listed will only be touched upon briefly, and some not at all. A common alternative is Hungeford, which achieves a comparable level of detail. Personally, I am partial to Isaacs. (Admittedly this probably stems from already being a big fan of his finite group theory book; still, I dig his writing style.) There are a couple topics (such as Hilbert's Theorem 90) which are not treated quite as thoroughly in Isaacs as they are in Lang, but for the most part, it is on the same level, and I find Isaacs exposition to be very clear and well-motivated.