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By Gert-Martin Greuel, Visit Amazon's Gerhard Pfister Page, search results, Learn about Author Central, Gerhard Pfister, , O. Bachmann, C. Lossen, H. Schönemann

ISBN-10: 3540735410

ISBN-13: 9783540735410

From the experiences of the 1st edition:

''It is unquestionably no exaggeration to assert that … a novel creation to Commutative Algebra goals to guide an extra degree within the computational revolution in commutative algebra … . one of the nice strengths and so much precise positive aspects … is a brand new, thoroughly unified remedy of the worldwide and native theories. … making it the most versatile and most productive structures of its type....another energy of Greuel and Pfister's ebook is its breadth of insurance of theoretical themes within the parts of commutative algebra closest to algebraic geometry, with algorithmic remedies of virtually each topic....Greuel and Pfister have written a particular and hugely beneficial e-book that are meant to be within the library of each commutative algebraist and algebraic geometer, specialist and amateur alike.''

J.B. Little, MAA, March 2004

The moment version is considerably enlarged by means of a bankruptcy on Groebner bases in non-commtative jewelry, a bankruptcy on attribute and triangular units with purposes to fundamental decomposition and polynomial fixing and an appendix on polynomial factorization together with factorization over algebraic box extensions and absolute factorization, within the uni- and multivariate case.

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Extra resources for A Singular Introduction to Commutative Algebra

Example text

Prove that the canonical map s A −→ A/Ij , a −→ (a + I1 , . . 14. Let K be a field and A a K–algebra. Then A is called an Artinian K–algebra if dimK (A) < ∞. Prove the following statements: (1) An Artinian K–algebra is Noetherian. (2) A is an Artinian K–algebra if and only if each descending chain of ideals I1 ⊃ I2 ⊃ I3 ⊃ . . ⊃ Ik ⊃ . . becomes stationary (that is, there exists some j0 such that Ij = Ij0 for all j ≥ j0 ). 15. 13. 8 (in characteristic 0), defining a minpoly. 16. Let f = x3 + y 3 + z 3 + 3xyz, and let I be the ideal in Q[x, y, z], respectively F3 [x, y, z], generated by f and its partial derivatives.

Moreover, the prime ideals in S −1 A are in one–to–one correspondence with the prime ideals in A which do not meet S. 6. (1) A P is multiplicatively closed for any prime ideal P ⊂ A. The localization of A with respect to A P is denoted by AP and AP = a a, b ∈ A, b ∈ P b is called the localization of A at the prime ideal P . The set a a ∈ P, b ∈ P P AP = b is clearly an ideal in AP . Any element a/b ∈ AP P AP satisfies a ∈ P , hence, b/a ∈ AP and, therefore, a/b is a unit. 3. In particular, if m ⊂ A is a maximal ideal then Am is local with maximal ideal mAm .

We have aλ aλ ∈ Iλ , aλ = 0 for almost all λ Iλ = λ∈Λ . λ∈Λ Because the empty sum is defined to be 0, the 0–ideal is generated by the empty set (but also by 0). The expression f = λ aλ fλ as a linear combination of the generators is, in general, by no means unique. For example, if I = f1 , f2 then we have the trivial relation f1 f2 − f2 f1 = 0, hence a1 f1 = a2 f2 with a1 = f2 , a2 = f1 . Usually there are also further relations, which lead to the notion of the module of syzygies (cf. Chapter 2).

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A Singular Introduction to Commutative Algebra by Gert-Martin Greuel, Visit Amazon's Gerhard Pfister Page, search results, Learn about Author Central, Gerhard Pfister, , O. Bachmann, C. Lossen, H. Schönemann

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