By Nathan Jacobson

A vintage textual content and traditional reference for a new release, this quantity and its significant other are the paintings of a professional algebraist who taught at Yale for 2 a long time. Nathan Jacobson's books own a conceptual and theoretical orientation, and also to their worth as lecture room texts, they function useful references.

Volume I explores the entire subject matters ordinarily lined in undergraduate classes, together with the rudiments of set thought, crew conception, earrings, modules, Galois idea, polynomials, linear algebra, and associative algebra. Its entire therapy extends to such rigorous themes as Lie and Jordan algebras, lattices, and Boolean algebras. routines seem during the textual content, besides insightful, rigorously defined proofs. Volume II includes all matters everyday to a first-year graduate path in algebra, and it revisits many issues from quantity I with larger intensity and class.

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Additional info for Basic Algebra I: Second Edition

Example text

The classical notation for this t is α(s). One calls this the image of s under α. In many books on algebra (including our previous ones) we find the notations sα and sα for α(s). This has advantages when we deal with the composite of maps. However, since the consensus clearly favors the classical notation α(s), we have decided to adopt it in this book. 32 Two maps are regarded as equal if and only if they have the same domain, the same co-domain and the same graphs. 3 If A is a subset of S, then we write α(A) = {α(a)|a ∈ A} and call this the image of A under α.

Moreover, the relation between E and π are reciprocal in the sense that πEπ = π and EπE = E. First, suppose E is given. If a ∈ S we let E (or simply ) = {b ∈ S|bEa}. We call E the equivalence class (relative to E or E-equivalence class) determined by a. In the first example considered in the last paragraph, the equivalence class E is the horizontal line through a and in the second, the equivalence class is the circle through a having center O: In both examples it is apparent that the set of equivalence classes is a partition of the plane.

For any infinite set there always exist bijective maps onto proper subsets. If S and T are finite sets then |S × T| = |S||T| and |ST| = |S||T| where ST is the set of maps of T into S. 64 An important result on cardinal numbers of infinite sets is the Schröder-Bernstein theorem: If we have injective maps of S into T and of T into S then |S| = |T|. 1 For a general reference book on set theory adequate for our purposes we refer the reader to the very attractive little book, Naive Set Theory. by Paul R.