By Serge Lang (auth.)
This booklet is meant as a simple textual content for a one-year direction in Algebra on the graduate point, or as an invaluable reference for mathematicians and pros who use higher-level algebra. It effectively addresses the fundamental suggestions of algebra. For the revised 3rd version, the writer has further workouts and made a number of corrections to the text.
Comments on Serge Lang's Algebra:
Lang's Algebra replaced the best way graduate algebra is taught, holding classical issues yet introducing language and methods of pondering from class conception and homological algebra. It has affected all next graduate-level algebra books.
April 1999 Notices of the AMS, asserting that the writer was presented the Leroy P. Steele Prize for Mathematical Exposition for his many arithmetic books.
The writer has a magnificent knack for proposing the $64000 and engaging principles of algebra in precisely the "right" approach, and he by no means will get slowed down within the dry formalism which pervades a few elements of algebra.
MathSciNet's assessment of the 1st edition
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Extra resources for Algebra
N - 2. For example to say that 0 -+ G' � G .!!. G" -+ 0 n ' 16 I, §3 G ROU PS is exact means that f is injective, that lm f = Ker g, and that g is surjective. If H = Ker g then this sequence is essentially the same as the exact sequence j 0 __.. H __.. j G __.. G/H j --+ 0. More precisely, there exists a commutative diagram G" 0 G' f G g 0 0 G G/H H in which the vertical maps are isomorphisms , and the rows are exact. Next we describe some homomorphisms, all of which are called canonical. (i) Let G, G' be groups and f: G __..
Hence B n C = 0. Let x E A. Since f(x) E A ' there exist integers n; , i E /, such that f(x) = L n i x� . ieI Applying f to x - L n i X ; , we find that this element lies in the kernel of f, iE say I x - L n; X ; = b E B. iei From this we see that x E B + C, and hence finally that A = B (f) C is a direct sum, as contended. 3. Let A be a free abelian group, and let B be a subgroup. Then B is also a free abelian group, and the cardinality of a basis of B is < the cardinality of a basis for A. Any two bases of B have the same cardinality.
Let A = Q/Z . Then Q/Z is a torsion abelian group , isomorphic to the direct sum of its subgroups (Q/Z)(p) . Each (Q/Z)(p) consists of those elements which can be represented by a rational number ajp k with a E Z and k some positive integer, i . e . a rational number having only a p-power in the denominator. See also Chapter IV , Theorem 5 . 1 . Example . In what follows we shall deal with finite abelian groups , so only a finite number of primes (dividing the order of the group) will come into play .
Algebra by Serge Lang (auth.)