Download Solutions to Atiyah and MacDonald's Introduction to by Papaioannou A. PDF

By Papaioannou A.

Show description

Read Online or Download Solutions to Atiyah and MacDonald's Introduction to Commutative Algebra PDF

Best algebra books

Groebner bases algorithm: an introduction

Groebner Bases is a method that gives algorithmic ideas to numerous difficulties in Commutative Algebra and Algebraic Geometry. during this introductory instructional the elemental algorithms in addition to their generalization for computing Groebner foundation of a collection of multivariate polynomials are provided.

The Racah-Wigner algebra in quantum theory

The improvement of the algebraic elements of angular momentum concept and the connection among angular momentum thought and unique subject matters in physics and arithmetic are coated during this quantity.

Wirtschaftsmathematik für Studium und Praxis 1: Lineare Algebra

Die "Wirtschaftsmathematik" ist eine Zusammenfassung der in den Wirtschaftswissenschaften gemeinhin benötigten mathematischen Kenntnisse. Lineare Algebra führt in die Vektor- und Matrizenrechnung ein, stellt Lineare Gleichungssysteme vor, berichtet über Determinanten und liefert Grundlagen der Eigenwerttheorie und Aussagen zur Definitheit von Matrizen.

Extra info for Solutions to Atiyah and MacDonald's Introduction to Commutative Algebra

Sample text

Assuming that JA = 0 and given any v ∈ B, we wish to construct a maximal ideal of B that doesn’t contain v. By applying exercise 21 to the ring Bv and its subring A (we use the usual embedding A −→ Bv and this is injective, because A is an integral domain) we obtain an s ∈ A − {0} such that, if Ω is an algebraically closed field and f : A −→ Ω doesn’t vanish at s, then f can be extended to a homomorphism B −→ Ω. Let m be a maximal ideal of A such that s ∈ / A, and let k = A/m be the residue field.

By renumbering the xi ’s, if necessary, we may assume that x1 , x2 , . . , xr are algebraically independent over k and each of the xr+1 , . . , xn are algebraic over k[x1 , x2 , . . , xr ]. Now we apply induction to the difference n − r; if n = r, then there is nothing to be shown, so assume that the proposition holds for n−1 generators and n > r. In this case, the generator xn is algebraic over k[x1 , . . , xn−1 ], hence there exists a polynomial f with coefficients in k such that f (x1 , x2 , .

C’) ⇒ (b’) This is the dual of the ’(c) ⇒ (a)’ statement above; we can thus prove it in exactly the same fashion, reversing the arrows. 11 Since f : A −→ B is flat, the induced map f ∗ : Spec(Bq ) −→ Spec(Aq ) is surjective (where q is a prime ideal of B and p is an ideal of A that lies over it in A), by chapter 3, exercise 18. Therefore, by exercise 10, f has the going-down property. 40 CHAPTER 5. 12 It’s obvious that A is integral over AG ; for, given a ∈ A, a is a root of the monic polynomial P (x) = G σ∈G (x − σ(a)).

Download PDF sample

Rated 4.34 of 5 – based on 26 votes