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The number of intermediate rings in FIP extension of integral domains. (English) Zbl 1451.13023
Authors’ abstract: Let $$R\subseteq S$$ be an extension of integral domains with only finitely many intermediate rings, where $$R$$ is not a field and $$S$$ is not necessarily the quotient field of $$R$$ or $$R$$ is not necessarily integrally closed in $$S$$. In this paper, we exactly determine the number of intermediate rings between $$R$$ and $$S$$ and give a way to compute it.
##### MSC:
 13B02 Extension theory of commutative rings 13B22 Integral closure of commutative rings and ideals 13E15 Commutative rings and modules of finite generation or presentation; number of generators 13E99 Chain conditions, finiteness conditions in commutative ring theory 13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations 13G05 Integral domains 13B30 Rings of fractions and localization for commutative rings
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