Download e-book for iPad: Algebra I: Basic Notions of Algebra (Encyclopaedia of by Igor R. Shafarevich, Aleksej I. Kostrikin, M. Reid

By Igor R. Shafarevich, Aleksej I. Kostrikin, M. Reid

ISBN-10: 0387170065

ISBN-13: 9780387170060

ISBN-10: 3540251774

ISBN-13: 9783540251774

ISBN-10: 3540264744

ISBN-13: 9783540264743

This e-book is wholeheartedly advised to each pupil or person of arithmetic. even though the writer modestly describes his booklet as 'merely an try to discuss' algebra, he succeeds in writing a very unique and hugely informative essay on algebra and its position in smooth arithmetic and technology. From the fields, commutative earrings and teams studied in each collage math path, via Lie teams and algebras to cohomology and type idea, the writer exhibits how the origins of every algebraic inspiration may be relating to makes an attempt to version phenomena in physics or in different branches of arithmetic. related fashionable with Hermann Weyl's evergreen essay The Classical teams, Shafarevich's new booklet is certain to turn into required analyzing for mathematicians, from newcomers to specialists.

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Extra resources for Algebra I: Basic Notions of Algebra (Encyclopaedia of Mathematical Sciences)

Example text

Let A be a graded ring; then A is Noetherian if and only if Ao is Noetherian and A is a ring of finite type over Ao. Proof. Obviously, the set of elements x e A for which x 0 = 0 in (1) is an ideal / 0 . It turns out that for the truth of the assertion in the theorem, it is sufficient for just this single ideal to be finitely generated. Indeed, we take a set of generators of/0, represent each generator in the form (1), and consider all the homogeneous terms x ; appearing in this way. ) which again obviously generate Io.

A wealth of material for making such choices is provided by taking E to be some finite field F,, and U to be a subspace of the vector space ¥£. Furthermore, the greatest success has been achieved by taking F9" and U to be finite-dimensional subspaces of the field ¥q(t) or even of F,(C), where C is an algebraic curve, and determining the choice of these subspaces by means of certain geometric conditions (such as considering functions with specified zeros and poles). Thus coding theory has turned out to be related to very delicate questions of algebraic geometry over finite fields.

A module of finite type over a PID is isomorphic to a direct sum of a finite number of cyclic modules. A cyclic module is either isomorphic to A or decomposes further as a direct sum of cyclic modules of the form A/(nk) where n is a prime element. The representation of a module as a direct sum of such modules is unique. If a module M is a torsion module then there are no summands isomorphic to A. This happens for example if A = Z and M is a finite Abelian group. In this case the theorem we have stated gives a classification of finite Abelian groups.

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Algebra I: Basic Notions of Algebra (Encyclopaedia of Mathematical Sciences) by Igor R. Shafarevich, Aleksej I. Kostrikin, M. Reid


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