id: 06252068
dt: b
an: 2014d.00684
au: Guin, Daniel
ti: Algebra. Volume 2. Rings, modules and multilinear algebra. L3, M1, M2.
(Algèbre. Tome 2. Anneaux, modules et algèbre multilinéaire. L3, M1,
M2.)
so: Collection Enseignement SUP. Mathématiques. Les Ulis: EDP Sciences (ISBN
978-2-7598-1001-7/pbk). xiv, 244~p. (2013).
py: 2013
pu: Les Ulis: EDP Sciences
la: FR
cc: H45 H65 H75
ut: commutative rings; ideals; factorial rings; polynomial rings; localization;
modules; Dedekind rings; tensor algebra; exterior algebra; derivations;
differential forms
ci: Zbl 1153.12001
li:
ab: The volume under review is the sequel to the French algebra textbook
“Algèbre I. Groupes, corps et théorie de Galois” [Les Ulis: EDP
Sciences. (2008; Zbl 1153.12001)] by the author and {\it T.
Hausberger}. Basically, the entire two-volume text is geared toward
upper-level undergraduates and graduate students in mathematics at
French universities. Also, it is meant to serve as a profound source
for those candidates preparing for the major civil-service examinations
CAPES or “L’agrégation” in France. While the first volume
provides a comprehensive introduction to the elements of group theory,
field theory and Galois theory, respectively, the current second
volume, this time with the author as the only author, is devoted to the
complementary topics of commutative ring theory, module theory, and
multilinear algebra. Accordingly, the material is organized in two
principal parts consisting of several chapters each. Within this
didactic disposition, Part I comprises the first seven chapters
covering the following topics successively: Chapter 1 gives an
introduction to the fundamental concepts of commutative ring theory
such as homomorphisms, ideals, prime and maximal ideals, products of
rings and the Chinese remainder theorem, the characteristic of a ring,
and quotient fields of integral domains. Chapter 2 discusses polynomial
rings, Euclidean rings, principal ideal domains, factorial rings, and
the notion of divisibility in rings. Chapter 3 continues the study of
polynomial rings, with particular emphasis on irreducible polynomials,
resultants and discriminants, derivations, and symmetric polynomials.
Chapter 4 turns to modules over commutative rings and their most
general properties, including direct sums and products of modules as
well as free modules and algebras in the course of the discussion, the
structure theorem for finitely generated modules over a principal ideal
theorem is the main topic of Chapter 5, while the subsequent Chapter 6
deals with some arithmetic concepts in ring theory. More precisely,
this chapter studies integral ring extensions, the notions of norm and
trace, Noetherian rings and modules, fractional ideals, Dedekind rings,
the norm of an ideal, the decomposition of prime ideals in ring
extensions, and the rudiments of ramification theory in this context.
Chapter 7 is devoted to the properties of the Hom-functor in module
theory and the related duality theory, with particular emphasis on
finitely generated free modules. Each of the Chapters 1‒7 concludes
with a section titled “Thèmes de réflexion”, where additional
topics are introduced through carefully guided exercises. Among the
themes inviting the reader to independent work are related topics such
as power series, Laurent series, further irreducibility criteria for
polynomials, universal properties of special module constructions, the
Jordan nomal form of a vector space endomorphism, prime numbers and
their ramification in number fields, injective and projective modules,
the infective hull, and modules of finite length. Part II of the
present book is titled “Multilinear algebra” and comprises the
remaining two chapters. Chapter 8 introduces tensor products of modules
and associative algebras, together with their basic (universal)
properties. The exercises to this chapter provide some basic facts on
flat and faithfully flat modules, respectively. Chapter 9 finally
treats alternating multilinear maps of modules, determinants of square
matrices over a ring, the exterior products of a finitely generated
free module, and the Grassmann algebra. The exercises acquaint the
reader with such related topics like derivations on a ring with values
in a module, algebraic differential forms, and vanishing properties of
exterior powers. The main text of the book is enhanced by two
appendices providing some general tools from set theory as used in the
course of the treatise. These appendices concern orderings and
well-orderings on sets, on the one hand, and cardinalities of infinite
sets on the other. Overall, the presentation of the material is
characterized by a high degree of clarity, rigor, and expository skill.
The wealth of accompanying exercises, illustrating examples, and
instructive remarks is another feature of this excellent textbook
which, besides, appears to be largely self-contained and pleasantly
versatile.
rv: Werner Kleinert (Berlin)