I have read, studied, or eprused many of Lang's books, and been at conferences enlivened by his presence. I believe he recentky died and was the subject of some appreciative articles in the AMS notices.
He was obviously a brilliant man, and even moreso, a marvel of energy, enthusiasm, and taste in mathematics. He wrote many of the books that raised the level of mathematics taught in college. Therse include especially his famous algebra book, still unmatched for its unabashed honesty about the expectations for a future professional. Langs book was essential as a grad student, to keep up with our courses.
He is also famous for his books on algebraic geometry, number theory, group representations (SL(2,R)), differential manifolds, as well as the undergraduate books on linear algebra, calculus, several variables, complex analysis, and even high school math.
He also lectured to high schools and high school teachers on what math is like to a researcher like himself, and also undertook several famous and lengthy crusades in political causes, which interest me less, but took lots of energy that woulod have been impossible for a less gifted person.Once at a conference we were all standing around the coffee machine gabbing, when time came to begin again. Serre sang out cheerily, "lets get back to work, or XXXXX (an NSF officer who was present) won't renew our grants!"
We all immediatey filed back in the auditorium laughing and chatting. That was my introduction to the personality of the irrepressible Serge Lang.
His books are famous for making difficult matters simple, but noit for featuring lots of the tedious exampels necessary to master them. In the professional world, and in his books, making and doing exercises is the responsibility of the reader.
His most famous exercise, and much more realistic than it may sound to a student today, was in his algebra book, where at the end of the section on homology he wrote: "Get hold of any book on homological algebra and prove all the theorems without looking at the proofs given in that book."
I have never met anyone else quite like Serge lang, nor have many people had his impact on graduate math education in my time.
His books are a good standard one should aspire to, but are widely considered too hard today, a sad statement about todays education standards. A few years ago, Hungerford's book was considered a compromise from lang's book, necessary because of the weakness of students. Today even Hungerford is considered too hard, and one often uses Dummit and Foote, an even lower level work than Hungerford.
I recommend using all three books, as Lang alone is insufficient for mastery, but those other books, as well as they do explain many things, and with their excellent exercises, still do not take one quite high enough in point of view, for professional success.
