Get Simple Extensions with the Minimum Degree Relations of PDF

By Susumu Oda

ISBN-10: 1584888512

ISBN-13: 9781584888512

Even if there are various forms of ring extensions, uncomplicated extensions haven't begun to be completely explored in a single e-book. protecting an understudied point of commutative algebra, basic Extensions with the minimal measure kin of critical domain names offers a entire remedy of assorted easy extensions and their houses. particularly, it examines numerous houses of easy ring extensions of Noetherian necessary domain names. As specialists who've been learning this box for over a decade, the authors current many arguments that they've constructed themselves, frequently exploring anti-integral, super-primitive, and ultra-primitive extensions. inside this framework, they research sure houses, reminiscent of flatness, integrality, and unramifiedness. the various themes mentioned comprise Sharma polynomials, vanishing issues, Noetherian domain names, denominator beliefs, unit teams, and polynomial jewelry. featuring a whole therapy of every subject, uncomplicated Extensions with the minimal measure kinfolk of imperative domain names serves as a fantastic source for graduate scholars and researchers all in favour of the realm of commutative algebra.

Show description

Read or Download Simple Extensions with the Minimum Degree Relations of Integral Domains PDF

Best algebra & trigonometry books

Kodaira-Spencer Maps in Local Algebra - download pdf or read online

The monograph contributes to Lech's inequality - a 30-year-old challenge of commutative algebra, originating within the paintings of Serre and Nagata, that relates the Hilbert functionality of the whole house of an algebraic or analytic deformation germ to the Hilbert functionality of the parameter house. A weakened model of Lech's inequality is proved utilizing a development that may be regarded as an area analog of the Kodaira-Spencer map identified from the deformation conception of compact complicated manifolds.

New PDF release: Noncommutative Geometry and Global Analysis: Conference in

This quantity represents the lawsuits of the convention on Noncommutative Geometric tools in international research, held in honor of Henri Moscovici, from June 29-July four, 2009, in Bonn, Germany. Henri Moscovici has made a couple of significant contributions to noncommutative geometry, international research, and illustration concept.

The recognition theorem for graded Lie algebras in prime by Georgia Benkart PDF

Quantity 197, quantity 920 (second of five numbers).

Extra resources for Simple Extensions with the Minimum Degree Relations of Integral Domains

Example text

We have αφ(J ) ⊆ αm ⊆ m ⊆ R for any α ∈ m : K m, and hence α ∈ R(J −1 ). Thus m : K m ⊆ R(J −1 ). But since R(J −1 ) ⊇ m :k m⊃ R, we − have R = R(J −1 )⊃ R, which is a contradiction. 3 Let J be a divisorial integral ideal of R. If R(J −1 ) = R, then there exists a nonzero element α ∈ K such that J = Iα . Proof We first show the following claim: If p ∈ Dp1 (R) contains J , then p is a prime divisor of J . 2. As p R p is a prime divisor of a principal ideal a R p = J p (cf. √ [Y]), p is a prime divisor of J .

Now we consider a certain over-ring of R, which is seen in Chapter 1. Let J be a fractional ideal of R. Recall R(J ) := J : K J, which is an over-ring of R. 9 Let J be a divisorial ideal of R. Then R(J ) = R if and only if R(J −1 ) = R. Proof Since J is divisorial, we have (J −1 )−1 = J. So it suffices to prove one of the implications. Assume that R(J ) = R. The implication R(J −1 ) ⊇ R is obvious. Take λ ∈ R(J −1 ). Then λJ −1 ⊆ J −1 . Thus R : K λJ −1 ⊇ R : K J −1 = (J −1 )−1 = J. On the other hand, we have R : λJ −1 = λ−1 R : K J −1 = λ−1 (R : K J −1 ) = λ−1 (J −1 )−1 = λ−1 J.

Let α be an element of an algebraic field extension L of the quotient field K of R and let π : R[X ] −→ R[α] be the R-algebra homomorphism sending X to α. Let ϕα (X ) be the monic minimal polynomial of α over K with deg ϕα (X ) = d d (R : R ηi ). For and write ϕα (X ) = X d + η1 X d−1 + · · · + ηd . Let I[α] := i=1 f (X ) ∈ R[X ], let c( f (X )) denote the ideal generated by the coefficients of f (X ). Let J[α] := I[α] c(ϕα (X )), which is an ideal of R and contains I[α] . The element α is called an anti-integral element of degree d over R if Ker(π ) = I[α] ϕα (X )R[X ].

Download PDF sample

Simple Extensions with the Minimum Degree Relations of Integral Domains by Susumu Oda

by Christopher

Rated 4.46 of 5 – based on 19 votes