New PDF release: Basic Commutative Algebra

By Balwant Singh

ISBN-10: 9814313610

ISBN-13: 9789814313612

This textbook, set for a one or semester path in commutative algebra, offers an creation to commutative algebra on the postgraduate and learn degrees. the most necessities are familiarity with teams, earrings and fields. Proofs are self-contained.

The e-book could be worthwhile to newbies and skilled researchers alike. the fabric is so prepared that the newbie can research via self-study or by means of attending a direction. For the skilled researcher, the e-book may possibly serve to give new views on a few famous effects, or as a reference.

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By a cyclic module, we mean a module generated by a single element. Thus a cyclic A-module is of the form As for some s. We denote by µ(M ) the least number of elements needed to generate a finitely generated A-module M. Note that M = 0 if and only if µ(M ) = 0, and M is cyclic if and only if µ(M ) ≤ 1. 5 Sums and Products. The sum of a family {Ni }i∈I of submodules of M , denoted i∈I Ni , is simply their sum as an additive subgroup. This is a submodule, the submodule generated by i∈I Ni , and it consists precisely of elements of the form j∈J xj with J a finite subset of I and xj ∈ Nj for every j ∈ J.

2) Immediate from (1). (3) The proof of this assertion is similar to that of (1) by noting that f must be given by f (x/s) = (1/s)f (x) for x ∈ M, s ∈ S. (4) This follows directly from the definition of the equivalence relation. (5) Immediate from (4). 2 Some Special Cases of Fractions. (1) Let S be the set of all nonzerodivisors of A. Then S is a multiplicative subset. In this case the ring S −1 A is called the total quotient ring of A. 1, whence A can be regarded as a subring of its total quotient ring.

Xn ) = α∈Nn where X α = X1α1 · · · Xnαn for α = (α1 , . . , αn ) ∈ Nn , fα ∈ A and fα = 0 for almost α, and that it is a commutative A-algebra with the following structure: fα X α = α∈Nn α∈Nn fα X α gα X α ⇔ fα = gα for every α ∈ Nn , gα X α + α∈Nn fα X α α∈Nn gα X α α∈Nn = α∈Nn α∈Nn fα X α a (fα + gα )X α , = α∈Nn α∈Nn   β+γ=α  fβ gγ  X α , afα X α , = α∈Nn for a ∈ A. The A-algebra structure corresponds to the ring homomorphism A → A[X1 , . . , Xn ] sending a ∈ A to the constant polynomial a, and this map identifies A as a subring of A[X1 , .

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Basic Commutative Algebra by Balwant Singh


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