I am taking abstract algebra in this semester. I find that my lecturer didn't provide enough examples for understanding. So I would like to ask what are the recommended books which has lots of solved problems and exercises? The lecturer said in this semester, he will cover group theory, ring theory and a bit of field theory. Which book contains extensive information in these three fields?
For a one-semester course in abstract algebra: undergraduate/first-course-level, I'd highly recommend Fraleigh's A First Course in Abstract Algebra. It's very well written, very readable, and includes a LOT of examples in the text itself, as well as in the exercises.
Personally, I've found the most success using this text for first-course/semester-length classes.
Another possible aid, in addition to a supplementary text, is Beachy's Abstract Algebra Online Study Guide, where you'll find an extensive study guide and practice problems. It may help to supplement areas where you're feeling weakest. I'll include here, as well, a link for accessing Beachy's ~150 page pdf Abstract Algebra - Study Guide for Beginners available for downloading, freely distributed.
Dummit & Foote has lots.
But following up on the "Artin" recommendation which is excellent, you can watch these great videos that follow "Artin" and see many examples very nicely articulated. You can probably get a lot more out of these lectures by Benedict Gross at Harvard:
I suggest you Algebra Through Practice collection by T. S. Blyth for first step and then Herstein or Fraleigh are good.
Michael Artin's book Algebra is very nice.
This is a difficult level to write for. I suggest you pick up two or three books from your library and try to read chapters in each concurrently.
Fraleigh is the book I got my start on in abstract algebra. I didn't like the book at the time- its exposition is very unclear at certain points- but now looking back I think it's not too bad. I'm using Gallian right now in my group theory class, which is also not too bad.
Dummit & Foote is not a book I would recommend as the sole textbook for a beginner, but it is great to have as supplementary reading. In particular, you are looking for examples, and D&F is all about examples. I would say the same thing about Artin.
I dislike Hungeford and Herstein, but this is just personal preference.
I was happy with Contemporary Abstract Algebra by Gallian when I worked through it (I used the 5th edition, but it's up to at least the 6th). It covers all three areas you mention, and has tons of exercises (at least 30-some in most of the main chapters, if not 50- or 60-some, plus supplementary exercises every few chapters) with solutions or hints for the odd-numbered ones. It also is heavy on applications, and has computer exercises, if you like that sort of thing. He puts real emphasis on the "contemporary" in the title.