LCM-stability and formal power series. (English) Zbl 1393.13042
It is well-known that the LCM-stability (an extension of integral domains \(A\subseteq B\) is called LCM-stable if for any couple \((a,b)\in A^2,\) \((aA\cap bA)B= aB\cap bB\)) of an extension \(A\subseteq B\) entails the LCM-stability of \(A[X]\subseteq B[X]\) (resp., \(A[\![X]\!]\subseteq B[\![X]\!]\)) when \(A\) is a locally GCD or Krull domain [H. Uda, Hiroshima Math. J. 13, 357–377 (1983; Zbl 0531.13001); Hiroshima Math. J. 18, No. 1, 47–52 (1988; Zbl 0683.13002)] (resp., when \(A\) is a Dedekind domain [J. T. Condo, Proc. Am. Math. Soc. 123, No. 8, 2333–2341 (1995; Zbl 0834.13017)]).
In the paper under review, the authors study whether the D-stability (resp., \(t\)-linkedness) of an extension \(A\subseteq B\) entails the D-stability (resp., \(t\)-linkedness) of \(A[X]\subseteq B[X]\) (resp., \(A[\![X]\!]\subseteq B[\![X]\!]\)). First, we need to collect some necessary notions to understand the content of the paper. Let \(A\subseteq B\) be an extension of integral domains. We say that the extension \(A\subseteq B\) is D-stable if for any divisorial ideal \(I\) of \(A\) we have \((IB)^{-1}=I^{-1}B\). Also, the extension \(A\subseteq B\) is said to be \(t\)-linked if for each finitely generated ideal \(I\) of \(A\) such that \(I^{-1}=A\) we have \((IB)^{-1}=B\). The authors show that if \(A\) is an integrally closed domain, then the extension \(A\subseteq B\) is D-stable if and only if \(A[X]\subseteq B[X]\) is D-stable. For the power series case, if \(A\) is a regular ring, then \(A\subseteq B\) is D-stable if and only if \(A[\![X]\!]\subseteq B[\![X]\!]\) is D-stable. Also, if \(A\subseteq B\) is an extension of Krull domains, then \(A[\![X]\!]\subseteq B[\![X]\!]\) is \(t\)-linked if and only if the extensions \(A\subseteq B\) and \(A[\![X]\!]_{A^*}\subseteq B[\![X]\!]_{B^*}\) are \(t\)-linked. Finally, the authors give an example of an LCM-stable extension \(A\subseteq B\) such that the extension \(A[\![X]\!]\subseteq B[\![X]\!]\) is not LCM-stable.
In the paper under review, the authors study whether the D-stability (resp., \(t\)-linkedness) of an extension \(A\subseteq B\) entails the D-stability (resp., \(t\)-linkedness) of \(A[X]\subseteq B[X]\) (resp., \(A[\![X]\!]\subseteq B[\![X]\!]\)). First, we need to collect some necessary notions to understand the content of the paper. Let \(A\subseteq B\) be an extension of integral domains. We say that the extension \(A\subseteq B\) is D-stable if for any divisorial ideal \(I\) of \(A\) we have \((IB)^{-1}=I^{-1}B\). Also, the extension \(A\subseteq B\) is said to be \(t\)-linked if for each finitely generated ideal \(I\) of \(A\) such that \(I^{-1}=A\) we have \((IB)^{-1}=B\). The authors show that if \(A\) is an integrally closed domain, then the extension \(A\subseteq B\) is D-stable if and only if \(A[X]\subseteq B[X]\) is D-stable. For the power series case, if \(A\) is a regular ring, then \(A\subseteq B\) is D-stable if and only if \(A[\![X]\!]\subseteq B[\![X]\!]\) is D-stable. Also, if \(A\subseteq B\) is an extension of Krull domains, then \(A[\![X]\!]\subseteq B[\![X]\!]\) is \(t\)-linked if and only if the extensions \(A\subseteq B\) and \(A[\![X]\!]_{A^*}\subseteq B[\![X]\!]_{B^*}\) are \(t\)-linked. Finally, the authors give an example of an LCM-stable extension \(A\subseteq B\) such that the extension \(A[\![X]\!]\subseteq B[\![X]\!]\) is not LCM-stable.
Reviewer: Ahmed Hamed (Monastir)
MSC:
| 13G05 | Integral domains |
| 13F25 | Formal power series rings |
| 13F05 | Dedekind, Prüfer, Krull and Mori rings and their generalizations |
| 13B99 | Commutative ring extensions and related topics |
| 13A15 | Ideals and multiplicative ideal theory in commutative rings |
Keywords:
commutative rings; formal power series; polynomial rings; LCM-stability; D-stability; t-linked; pvmd; Krull ringsReferences:
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