A directed -group that is not a group of divisibility
A General Theory of Almost Factoriality.
A note on generalized primary rings
A note on projective modules and multiplication modules
A note on the dual of a finitely generated multiplication module
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....
Cale Bases in Algebraic Orders
Let be a non-maximal order in a finite algebraic number field with integral closure . Although is not a unique factorization domain, we obtain a positive integer and a family (called a Cale basis) of primary irreducible elements of such that has a unique factorization into elements of for each coprime with the conductor of . Moreover, this property holds for each nonzero when the natural map is bijective. This last condition is actually equivalent to several properties linked...
Chains of factorizations in orders of global fields
Chains of factorizations in weakly Krull domains
Class Groups of Localities of Rings of Invariants of Reductive Algebraic Groups.
Cohen-Macaulayness of multiplication rings and modules
Let R be a commutative multiplication ring and let N be a non-zero finitely generated multiplication R-module. We characterize certain prime submodules of N. Also, we show that N is Cohen-Macaulay whenever R is Noetherian.
Congruence monoids
Convex-compact sets and Banach discs
Every relatively convex-compact convex subset of a locally convex space is contained in a Banach disc. Moreover, an upper bound for the class of sets which are contained in a Banach disc is presented. If the topological dual of a locally convex space is the -closure of the union of countably many -relatively countably compacts sets, then every weakly (relatively) convex-compact set is weakly (relatively) compact.
Direct method for primary decomposition.
Diversity in inside factorial monoids
In a recent paper (Diversity in Monoids, Czech. Math. J. 62 (2012), 795–809), the last two authors introduced and developed the monoid invariant “diversity” and related properties “homogeneity” and “strong homogeneity”. We investigate these properties within the context of inside factorial monoids, in which the diversity of an element counts the number of its different almost primary components. Inside factorial monoids are characterized via diversity and strong homogeneity. A new invariant complementary...
Diversity in monoids
Let be a (commutative cancellative) monoid. A nonunit element is called almost primary if for all , implies that there exists such that or . We introduce a new monoid invariant, diversity, which generalizes this almost primary property. This invariant is developed and contextualized with other monoid invariants. It naturally leads to two additional properties (homogeneity and strong homogeneity) that measure how far an almost primary element is from being primary. Finally, as an application...
Divisors in a Dedekind domain
Elasticity in certain block monoids via the Euclidean table