It is well known that an integral domain is a valuation domain if and only if it possesses only one finitary ideal system (Lorenzen -system of finite character). We prove an analogous result for root-closed (cancellative) monoids and apply it to give several new characterizations of Prüfer (multiplication) monoids and integral domains.
The tropical semifield, i.e., the real numbers enhanced by the operations of addition and maximum, serves as a base of tropical mathematics. Addition is an abelian group operation, whereas the maximum defines an idempotent semigroup structure. We address the question of the geometry of idempotent semigroups, in particular, tropical algebraic sets carrying the structure of a commutative idempotent semigroup. We show that commutative idempotent semigroups are contractible, that systems of tropical...
In this paper, we extend some results of D. Dolzan on finite rings to profinite rings, a complete classification of profinite commutative rings with a monothetic group of units is given. We also prove the metrizability of commutative profinite rings with monothetic group of units and without nonzero Boolean ideals. Using a property of Mersenne numbers, we construct a family of power commutative non-isomorphic profinite semiprimitive rings with monothetic group of units.
Cet article est le premier d’une série de trois articles consacrés aux images directes d’isocristaux : ici nous considérons des isocristaux sans structure de Frobenius ; dans le deuxième [Et 6] (resp. le troisième [Et 7]), nous introduirons une structure de Frobenius dans le contexte convergent (resp. surconvergent).Pour un morphisme propre et lisse relevable nous établissons la surconvergence des images directes, grâce à un théorème de changement de base pour un morphisme propre entre espaces rigides...
We describe all those indecomposable primarily comultiplication modules with finite-dimensional top over pullback of two Dedekind domains. We extend the definition and results given by R. Ebrahimi Atani and S. Ebrahimi Atani [Algebra Discrete Math. 2009] to a more general primarily comultiplication modules case.