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Algebra. 3. ed. (English) Zbl 0848.13001
Summary: The first edition of this standard textbook appeared 1965 (Zbl 0193.34701). – The author writes in the foreword of the second edition (1984): “In this second edition, I have added several topics, having mostly to do with commutative algebra and homological algebra, for instance: projective and injective modules, leading to an extended treatment of homological algebra, with derived functors, the Hilbert syzygy theorem, and a more thorough discussion of $K$-groups and Euler characteristics; the Quillen-Suslin theorem (previously Serre’s conjecture) that finite projective modules over a polynomial ring are free; the Weierstrass preparation theorem; the Hilbert polynomial in connection with filtered and graded modules; more material on tensor products, like flat modules and derivations; etc... – I have added a number of new exercises, especially in the chapters which are most likely to be of fundamental use, like the chapter on group theory, Noetherian rings, Galois theory, and tensor products.” In the foreword of the present third edition the author adds: “After almost a decade since the second edition. I find that the basic topics of algebra have become stable, with one exception. I have added two sections on elimination theory, complementing the existing section on the resultant. Algebraic geometry having progressed in many ways, it is now sometimes returning to older and harder problems, such as searching for the effective construction of polynomials vanishing on certain algebraic sets, and the older elimination procedures of last century serve as an introduction to those problems.