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Arithmetical theory of monoid homomorphisms. (English) Zbl 0805.20057

This paper gives a careful unified theory of certain homomorphisms, called divisor homomorphisms and essential homomorphisms, and families of such homomorphisms between commutative cancellative monoids. These homomorphisms generalize divisor theories for monoids. Some of the topics investigated include Krull monoids and their generalizations, the integral closure of a monoid, finiteness results and chain conditions for monoids, and applications to integral domains. These concepts generalize well known arithmetic (or divisibility) properties for integral domains and correspond to when an integral domain is an intersection of localizations or special types of valuation overrings.

MSC:

20M15 Mappings of semigroups
20M14 Commutative semigroups
13G05 Integral domains
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References:

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