## $$T$$-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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