A Singular Introduction to Commutative Algebra by Gert-Martin Greuel, Visit Amazon's Gerhard Pfister Page,

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

From the experiences of the 1st edition:

''It is unquestionably no exaggeration to assert that … a novel advent to Commutative Algebra goals to steer yet another degree within the computational revolution in commutative algebra … . one of the nice strengths and so much certain gains … is a brand new, thoroughly unified remedy of the worldwide and native theories. … making it some of the most versatile and most productive platforms of its type....another power of Greuel and Pfister's booklet is its breadth of assurance of theoretical subject matters 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 worthy booklet that are meant to be within the library of each commutative algebraist and algebraic geometer, professional and beginner 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 functions to basic 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

Ym ] and consider the product ordering > = (>1 , >2 ) on Mon(x1 , . . , xn , y1 , . . , ym ), where >1 is global on Mon(x1 , . . , xn ) and >2 is local on Mon(y1 , . . , ym ). Then xα y γ > 1 > y β for all α, β = 0, all γ and hence S> = K ∗ + y · K[y]. It follows that K[x, y]> = (K[y] y )[x] , which equals K[y] y ⊗K K[x] (cf. 7 for the tensor product). (2) Now let >1 be local and >2 global, > = (>1 , >2 ), then xα y γ < 1 < y β for all α, β = 0, all γ and hence S> = K ∗ + x K[x, y]. We obtain strict inclusions (K[x] x )[y] K[x, y]> K[x, y] x , since 1/(1 + xy) is in the second but not in the first and 1/y is in the third but not in the second ring.

1) I is a prime ideal if I = A and if for each a, b ∈ A : ab ∈ I ⇒ a ∈ I or b ∈ I. (2) I is a maximal ideal if I = A and if it is maximal with respect to inclusion A and I ⊂ I implies I = I ). (that is, for any ideal I (3) The set of prime ideals is denoted by Spec(A) and the set of maximal ideals by Max(A). The set of prime ideals Spec(A) of a ring A is made a topological space by endowing it with the so–called Zariski topology, creating, thus, a bridge between algebra and topology. 3, for a short introduction.

Fk , . . of minimal possible degree. If di = deg(fi ), fi = ai xdi + lower terms in x , then d1 ≤ d2 ≤ . . and a1 ⊂ a1 , a2 ⊂ . . is an ascending chain of ideals in A. By assumption it is stationary, that is, a1 , . . , ak = a1 , . . , ak+1 for some k, hence, ak+1 = ki=1 bi ai for suitable bi ∈ A. Consider the polynomial k k bi xdk+1 −di fi = ak+1 xdk+1 − g = fk+1 − i=1 bi ai xdk+1 + lower terms . i=1 Since fk+1 ∈ I f1 , . . , fk , it follows that g ∈ I f1 , . . , fk is a polynomial of degree smaller than dk+1 , a contradiction to the choice of fk+1 .

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