The catenarian property of power series rings over a Prüfer domain.(English)Zbl 0577.13010

The author has defined previously an ideal A of a commutative ring R to be of strong finite type provided there is a finitely generated ideal B of R with $$B\subseteq A$$ and a positive integer k such that $$a^ k\in B$$ for each $$a\in A$$; the ring R is said to have the SFT-property provided that each ideal of R is of strong finite type. This paper is concerned with a Prüfer domain D that has the SFT-property: the main results are that the power series ring D[[X]] is catenarian, but that if $$n>1$$ and dim D$$>1$$ then the power series ring $$D[[X_ 1,...,X_ n]]$$ is not catenarian.
Reviewer: R.Y.Sharp

MSC:

 13E99 Chain conditions, finiteness conditions in commutative ring theory 13F25 Formal power series rings 13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations 13C15 Dimension theory, depth, related commutative rings (catenary, etc.) 13E15 Commutative rings and modules of finite generation or presentation; number of generators
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