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\(T\)-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.

MSC:

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
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