@Kevin Lin: At least in Algebraic Geometry, the surprising fact is that $\mathrm{hom}(\mathrm{Spec}(A), \mathrm{Spec}(\mathbb{Z}[x]))$ carries all the information of $\mathrm{Spec}(A)$. So, in this case, we only need to consider ONE set of maps from our object to $\mathrm{Spec}(\mathbb{Z}[x])$.

Thanks for your answer. This is exactly the question I am asking: when we write $X = \mathrm{Spec} A$, already implicitly, we are viewing $A$ as the ring of functions from $X$ to $\mathrm{Spec} K[x]$.

@José Figueroa-O'Farrill: The question is not intended to be misleading. If I had known everything about it, I wouldn't ask the question anyway.

@Andrew L: Thanks a lot for your comments! I definitely need to look up Jacobson's book since many people talk about it. @Max: I hope you've been reading our comments. Since I'm only an undergraduate, one implication is that I have no idea about how to teach so that people can learn effectively. That means you would probably be much better off listening to what Andrew L has to say. The list above is more of my personal list so far and all I know is that I enjoyed working with those books. Good luck!

Lang's definitely hard for a beginner, including myself. I struggled hard with it, but it's worth the effort. The way everything formulated is amazing, especially the part on Galois Theory. I read the book for my first abstract algebra course (which is only 2 years ago), and it was hard. But in the end, I got prepared. I think "having pain" this way is a good way to learn. ForCA, reading and doing all exercises in Atiyah Macdonald is good since it's not too long . Eisenbud's book is definitely good, if one has the time to read through it (admittedly, I only looked here and there in the book).

Thanks for your comment! I like the books by M. Lee too, even though I haven't really gone through them. For the Analysis book, are you talking about Rudin's Principles of Mathematical Analysis? I really liked this book when I took the first course in Analysis. For the other books, I haven't read them, so I don't know.