A General Theory of Almost Factoriality.
A note on projective modules and multiplication modules
A simple characterization of principal ideal domains
1. Introduction. In this note we give necessary and sufficient conditions for an integral domain to be a principal ideal domain. Curiously, these conditions are similar to those that characterize Euclidean domains. In Section 2 we establish notation, discuss related results and prove our theorem. Finally, in Section 3 we give two nontrivial applications to real quadratic number fields.
A weakening of the euclidean property for integral domains and applications to algebraic number theory. II.
A weakening of the euclidean property for integral domains and applications to algebraic number theory. I.
An addendum to the paper: “Arithmetic functions over rings with zero divisors” by Ruangsinsap, Laohakosol and P. Udomkavanich.
Arithmetic functions over rings with zero divisors.
Arithmetic of non-principal orders in algebraic number fields
Let be an order in an algebraic number field. If is a principal order, then many explicit results on its arithmetic are available. Among others, is half-factorial if and only if the class group of has at most two elements. Much less is known for non-principal orders. Using a new semigroup theoretical approach, we study half-factoriality and further arithmetical properties for non-principal orders in algebraic number fields.
Arithmetic progressions in a unique factorization domain
Atomicity and the fixed divisor in certain pullback constructions
Let D be an integral domain with field of fractions K. In this article, we use a certain pullback construction in the spirit of Int(E,D) that furnishes many examples of domains between D[x] and K[x] in which there are elements that do not admit a finite factorization into irreducible elements. We also define the notion of a fixed divisor for this pullback construction to characterize all of its irreducible elements and those nonzero nonunits that do admit a finite factorization into irreducibles....
Boundary map and overrings of half-factorial domains
We investigate factorization of elements in overrings of a half-factorial domain in relation with the behaviour of the boundary map of . It turns out that a condition, called , on the extension plays a central role in this study. We finally apply our results to the special case of polynomial rings.
Class Groups of Localities of Rings of Invariants of Reductive Algebraic Groups.
Complete local factorial rings which are not Cohen-Macaulay in characteristic
Conjugated and symmetric polynomial equations. II. Discrete-time systems
Division et composition dans l'anneau des séries de Dirichlet analytiques
Ce travail est une étude analytique locale de l’anneau des séries de Dirichlet convergentes. Dans un premier temps, on établit des propriétés arithmétiques de cet anneau ; on prouve en particulier sa factorialité, que l’on déduit de théorèmes de division du type Weierstrass. Ensuite, on s’intéresse à des problèmes de composition. Soient et des séries de Dirichlet convergentes. On sait que avec est encore une série de Dirichlet convergente. On étudie la réciproque : sous les hypothèses que...
Divisors on Varieties of Complexes.
Elasticity of A + XB[X] when A ⊂ B is a minimal extension of integral domains
We investigate the elasticity of atomic domains of the form ℜ = A + XB[X], where X is an indeterminate, A is a local domain that is not a field, and A ⊂ B is a minimal extension of integral domains. We provide the exact value of the elasticity of ℜ in all cases depending the position of the maximal ideals of B. Then we investigate when such domains are half-factorial domains.
Equivalent Conditions for Unique Factorization
Essentially indecomposable modules over a complete discrete valuation ring
Exponential Automorphisms of Complete Local Rings.