Related papers: On transfer Krull monoids

Every Krull monoid has a transfer homomorphism onto a monoid of zero-sum sequences over a subset of its class group. This transfer homomorphism is a crucial tool for studying the arithmetic of Krull monoids. In the present paper, we…

We study the arithmetic of monoids of regular elements of commutative rings with zero-divisors. Our focus is on Krull rings and on some of their generalizations (such as weakly Krull rings and C-rings). We establish sufficient conditions…

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…

Transfer Krull monoids are a recent concept including all commutative Krull domains and also, for example, wide classes of non-commutative Dedekind domains. We show that transfer Krull monoids are fully elastic (i.e., every rational number…

We introduce a new invariant describing the structure of sets of lengths in atomic monoids and domains. For an atomic monoid $H$, let $\Delta_{\rho} (H)$ be the set of all positive integers $d$ which occur as differences of arbitrarily long…

Let $H$ be a transfer Krull monoid over a finite ablian group $G$ (for example, rings of integers, holomorphy rings in algebraic function fields, and regular congruence monoids in these domains). Then each nonunit $a \in H$ can be written…

We establish a characterization (under some natural conditions) of those orders in Dedekind domains which allow a transfer homomorphism to a monoid of zero-sum sequences. As a consequence, the inclusion map to the Dedekind domain is a…

There are several families of atomic monoids whose arithmetical invariants have received a great deal of attention during the last two decades. The factorization theory of finitely generated monoids, strongly primary monoids, Krull monoids,…

We carry out a detailed investigation of congruence half-factorial Krull monoids with finite cyclic class group and related problems. Specifically, we determine precisely all relatively large values that can occur as a minimal distance of a…

Let $R$ be a (possibly noncommutative) ring and let $\mathcal C$ be a class of finitely generated (right) $R$-modules which is closed under finite direct sums, direct summands, and isomorphisms. Then the set $\mathcal V (\mathcal C)$ of…

For an element $a$ of a monoid $H$, its set of lengths $\mathsf L (a) \subset \mathbb N$ is the set of all positive integers $k$ for which there is a factorization $a=u_1 \cdot \ldots \cdot u_k$ into $k$ atoms. We study the system $\mathcal…

It is well-known that a C-monoid is completely integrally closed if and only if its reduced class semigroup is a group and if this holds, then the C-monoid is a Krull monoid and the reduced class semigroup coincides with the usual class…

An atomic monoid is length-factorial if each two distinct factorizations of any element have distinct factorization lengths. We provide a characterization of length-factorial Krull monoids in terms of their class groups and the distribution…

Let D be a Krull domain and Int(D) the ring of integer-valued polynomials on D. For any f in Int(D), we explicitly construct a divisor homomorphism from [f], the divisor-closed submonoid of Int(D) generated by f, to a finite sum of copies…

Let $H$ and $L$ be quantum groupoids. If $H$ has a quasitriangular structure, then we show that $L$ induces a Hopf algebra $C_{L}(L_s)$ in the category $_{H}\mathcal{M}$, which generalizes the transmutation theory introduced by Majid.…

We study the algebraic and arithmetic structure of monoids of invertible ideals (more precisely, of $r$-invertible $r$-ideals for certain ideal systems $r$) of Krull and weakly Krull Mori domains. We also investigate monoids of all nonzero…

We study the arithmetic of seminormal $v$-noetherian weakly Krull monoids with nontrivial conductor which have finite class group and prime divisors in all classes. These monoids include seminormal orders in holomorphy rings in global…

This is a survey on factorization theory. We discuss finitely generated monoids (including affine monoids), primary monoids (including numerical monoids), power sets with set addition, Krull monoids and their various generalizations, and…

We develop a new technique for studying monomial ideals in the standard polynomial rings $A[X_1,\ldots,X_d]$ where $A$ is a commutative ring with identity. The main idea is to consider induced ideals in the semigroup ring…

Let $H$ be a Krull monoid with class group $G$ and suppose that each class contains a prime divisor. Then every element $a \in H$ has a factorization into irreducible elements, and the set $\mathsf L (a)$ of all possible factorization…