On the First Galois Cohomology Group of the Algebraic Group SL1(D)

Let A be a central simple algebra over a field F. Denote the reduced norm of A over F by Nrd: A* → F* and its kernel by SL₁(A). For a field extension K of F, we study the first Galois Cohomology group H ¹(K,SL₁(A)) by two methods, valuation theory for division algebras and K-theory. We shall show that this group fails to be stable under purely transcendental extension and formal Laurent series.     

The Ordered Group of Invertible Ideals of a Prüfer Domain of Finite Character

Abstract

Let D be a Prüfer domain, and denote by ± b?(D) the multiplicative group of all invertible fractional ideals of D, ordered by A ≤ B if and only if A ? B. Denote by G _i the value group of the valuation associated with the valuation ring D _{M i} , where {M _i } _i∈I is the collection of all maximal ideals of D. In this note we prove that the natural map from ± b?(D) into ± b∏ _i∈I G _i is an isomorphism onto the cardinal sum ± b? _i∈I G _i if and only if D is h-local. As a corollary, the group of divisibility of an h-local Bézout domain is isomorphic to ± b? _i∈I G _i , the notation being as above.     

Valuations on a Module

The purpose of this article is to introduce the notion of valuations on a module over a commutative ring. Some basic results on valuations on a module are established. Moreover, the notion of valuation triples of a module is also introduced. In this article, it is shown that the so-called valuation triples correspond to valuations on a module.     

Rings Over which the Krull Dimension and the Noetherian Dimension of All Modules Coincide

We denote by 𝒜(R) the class of all Artinian R-modules and by 𝒩(R) the class of all Noetherian R-modules. It is shown that 𝒜(R) ? 𝒩(R) (𝒩(R) ? 𝒜(R)) if and only if 𝒜(R/P) ? 𝒩(R/P) (𝒩(R/P) ? 𝒜(R/P)), for all centrally prime ideals P (i.e., ab ∈ P, a or b in the center of R, then a ∈ P or b ∈ P). Equivalently, if and only if 𝒜(R/P) ? 𝒩(R/P) (𝒩(R/P) ? 𝒜(R/P)) for all normal prime ideals P of R (i.e., ab ∈ P, a, b normalize R, then a ∈ P or b ∈ P). We observe that finitely embedded modules and Artinian modules coincide over Noetherian duo rings. Consequently, 𝒜(R) ? 𝒩(R) implies that 𝒩(R) = 𝒜(R), where R is a duo ring. For a ring R, we prove that 𝒩(R) = 𝒜(R) if and only if the coincidence in the title occurs. Finally, if Q is the quotient field of a discrete valuation domain R, it is shown that Q is the only R-module which is both α-atomic and β-critical for some ordinals α,β ≥ 1 and in fact α = β = 1.     

Kronecker Function Rings of Transcendental Field Extensions

We consider the ring Kr(F/D), where D is a subring of a field F, that is the intersection of the trivial extensions to F(X) of the valuation rings of the Zariski–Riemann space consisting of all valuation rings of the extension F/D and investigate the ideal structure of Kr(F/D) in the case where D is an affine algebra over a subfield K of F and the extension F/K has countably infinite transcendence degree, by using the topological structure of the Zariski–Riemann space. We show that for any pair of nonnegative integers d and h, there are infinitely many prime ideals of dimension d and height h that are minimal over any proper nonzero finitely generated ideal of Kr(F/D).     

Primes and Right Ideals in Right Cones

A right cone H in a group G is a submonoid of G that generates G and aH ? bH for a, b ? H implies bH ? aH. With any right ideal I ≠ H of H a completely prime ideal P _r (I) of H is associated and the set 𝒫(I) of right ideals I′ of H with the same associated prime ideal P′ =P _r (I) is determined if P′·? P″ is a right invariant segment in H. The set 𝒫(I) is also described if P _r (I) is a limit prime.     

Dominance and Transmissions in Supertropical Valuation Theory

This article is a sequel of [4 Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2011 ). Supertropical semirings and supervaluations . J. Pure and Applied Alg. 215 ( 10 ): 2431 – 2463 .[Crossref], [Web of Science ®] , [Google Scholar]], where we defined supervaluations on a commutative semiring R and studied a dominance relation ? ≥ ψ between supervaluations ? and ψ on R, aiming at an enrichment of the algebraic tool box for use in tropical geometry.

A supervaluation ?: R → U is a multiplicative map from R to a supertropical semiring U, cf. [4 Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2011 ). Supertropical semirings and supervaluations . J. Pure and Applied Alg. 215 ( 10 ): 2431 – 2463 .[Crossref], [Web of Science ®] , [Google Scholar]], [7 Izhakian , Z. , Rowen , L. ( 2011 ). Supertropical matrix algebra . Israel J. Math. 182 ( 1 ): 383 – 424 .[Crossref], [Web of Science ®] , [Google Scholar]], [8 Izhakian , Z. , Rowen , L. ( 2010 ). Supertropical polynomials and resultants . J. Alg. 324 : 1860 – 1886 . (Preprint at arXiv:0902.2155.) [Crossref], [Web of Science ®] , [Google Scholar]], [5 Izhakian , Z. , Knebusch , M. , Rowen , L. Supertropical monoids: Basics and canonical factorization . Preprint at arXiv:1108.1880 . [Google Scholar]], [9 Maclane , S. ( 1998 ). Categories for the Working Mathemtician. , 4th ed. Springer Vereag . [Google Scholar]], with further properties, which mean that ? is a sort of refinement, or covering, of an m-valuation (= monoid valuation) v: R → M. In the most important case, that R is a ring, m-valuations constitute a mild generalization of valuations in the sense of Bourbaki [1 Bourbaki , N. Algèbre Commutative VI, §3 No. 1 . [Google Scholar]], while ? ≥ ψ means that ψ: R → V is a sort of coarsening of the supervaluation ?. If ?(R) generates the semiring U, then ? ≥ ψ iff there exists a “transmission” α: U → V with ψ = α ○ ?.

Transmissions are multiplicative maps with further properties, cf. [4 Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2011 ). Supertropical semirings and supervaluations . J. Pure and Applied Alg. 215 ( 10 ): 2431 – 2463 .[Crossref], [Web of Science ®] , [Google Scholar], Section 5]. Every semiring homomorphism α: U → V is a transmission, but there are others which lack additivity, and this causes a major difficulty. In the main body of the article we study surjective transmissions via equivalence relations on supertropical semirings. We put special emphasis on homomorphic equivalence relations. Even those are often much more complicated than congruences by ideals in usual commutative algebra.     

Power-Central Elements in Tensor Products of Symbol Algebras

Let A be a central simple algebra over a field F. Let k₁,…, k_r be cyclic extensions of F such that k₁ ?_F… ?_Fk_r is a field. We investigate conditions under which A is a tensor product of symbol algebras where each field k_i lies in a symbol F-algebra factor of the same degree as k_i over F. As an application, we give an example of an indecomposable algebra of degree 8 and exponent 2 over a field of 2-cohomological dimension 4.     

Multi-stage IT project evaluation: The flexibility value obtained by implementing and resolving Berk, Green and Naik (2004) model

In this paper, we evaluate a multi-stage information technology investment project, by implementing and resolving Berk, Green and Naik’s (2004) model, which takes into account specific features of IT projects and considers the real option to suspend investment at each stage. We present a particular case of the model where the project value is the solution of an optimal control problem with a single state variable. In this case, the model is more intuitive and tractable. The case study confirms the practical potential of the model and highlights the importance of the real-option approach compared to classical discounted cash flow techniques in the valuation of IT projects.