Krull dimension measures the depth of the spectrum Spec(R) of a commutative ring R. Since Spec(R) is a spectral space, Krull dimension can be defined for spectral spaces. Utilizing Stone duality, it can also be defined for distributive lattices. For an arbitrary topological space, the notion of Krull dimension is less useful. Isbell [23] remedied this by introducing the concept of graduated dim...

A (not necessarily commutative) Krull monoid—as introduced by Wauters—is defined as a completely integrally closed monoid satisfying the ascending chain condition on divisorial two-sided ideals. We study the structure of these Krull monoids, both with ideal theoretic and with divisor theoretic methods. Among others we characterize normalizing Krull monoids by divisor theories. Based on these re...

We develop the theory of Krull dimension for S4-algebras and Heyting algebras. This leads to the concept of modal Krull dimension for topological spaces. We compare modal Krull dimension to other well-known dimension functions, and show that it can detect differences between topological spaces that Krull dimension is unable to detect. We prove that for a T1-space to have a finite modal Krull di...

An additive category in which each object has a Krull-Remak-Schmidt decomposition—that is, finite direct sum decomposition consisting of objects with local endomorphism rings—is known as Krull-Schmidt category. A Hom-finite is an for there commutative unital ring k, such that Hom-set length k-module. The aim this note to provide proof Krull-Schmidt, if and only it split idempotents, indecomposa...

The main aim of this paper is to show that an AB5 module whose small submodules have Krull dimension has a radical having Krull dimension. The proof uses the notion of dual Goldie dimension.

This paper determines when the Krull-Schmidt property holds for all finitely generated modules and for maximal Cohen-Macaulay modules over one-dimensional local rings with finite Cohen-Macaulay type. We classify all maximal CohenMacaulay modules over these rings, beginning with the complete rings where the Krull-Schmidt property is known to hold. We are then able to determine when the Krull-Sch...

1 Introduction

Let D be a domain and let S torsion-free monoid such that has characteristic 0 or the quotient group of satisfies ascending chain condition on cyclic subgroups. We give characterization when algebra [ ] is weakly Krull. As corollaries, we reobtain results Krull resp. factorial, due to Chouinard Chang. Furthermore, deduce generalized algebras analogous our main result characterize domains among ...

Let T be the set of minimal primes of a Krull domain A. If S is a subset of T9 we form B = n AP for PeS and study the relation of the class group of B to that of A. We find that the class group of B is always a homomorphic image of that of A. We use this type of construction to obtain a Krull domain with specified class group and then alter such a Krull domain to obtain a Dedekind domain with t...

The concepts of tropical semiring and tropical hypersurface, are extended to the case of an arbitrary ordered group. Then, we define the tropicalization of a polynomial with coefficients in a Krull-valued field. After a close study of the properties of the operator “tropicalization” we conclude with an extension of Kapranov’s theorem to algebraically closed fields together with a valuation over...