\(T\)-block monoids and their arithmetical applications to certain integral domains. (English) Zbl 0809.13013

The author develops a combinatorial machinery based on the notion of \(T\)- block monoid in order to tackle arithmetical problems for certain integral domains. – In an abelian group \(G\), the author considers the free abelian monoid on a basis \(G_ 0 \subset G\), a monoid \(T \subset G\) and the monoid \({\mathcal F} (G_ 0) \times T\) (the elements of the form \(\prod_{x \in G_ 0} x^{n_ x} \cdot t\), \(n_ x \in \mathbb{N}\) and \(n_ x = 0\) for all but finitely many \(x \in G_ 0\), \(t \in T)\). If a monoid homomorphism \(\iota : {\mathcal F} (G_ 0) \times T \to G\) is content, this means that \(\iota (g) = g\) for all \(g \in G_ 0\), then \({\mathcal B} (G_ 0, T, \iota) : = \text{Ker} (\iota) \subset {\mathcal F} (G_ 0) \times T\) is called a \(T\)-block monoid over \(G_ 0\).
First some algebraic properties of \(T\)-block monoids are studied in the paper, then a \(T\)-block monoid \({\mathcal B}\) is attached to a monoid \(S\) admitting a \(T\)-divisor homomorphism and it is shown that all questions concerning lengths of factorizations in \(S\) can be studied in \({\mathcal B}\). Some structural results and their applications to concrete integral domains (for example to generalized Cohen-Kaplansky domains and to \(\mathbb{Z}\)-rings) are also given in the paper.


13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
20M25 Semigroup rings, multiplicative semigroups of rings
13F15 Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
14C22 Picard groups
Full Text: DOI


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