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Coherentlike conditions in pullbacks. (English) Zbl 0896.13007

Let \(M\) be a (nonzero) maximal ideal of a domain \(T\), let \(k=T/M\) be the residue field, let \(\varphi: T\to k\) be the natural projection, and let \(D\) be a proper subring of \(k\). Let \(R= \varphi^{-1} (D)\) be the domain arising from the following pullback of canonical homomorphisms: \[ \begin{matrix} R & \to & D \\ \downarrow & & \downarrow \\ T & @>\varphi>> & k & =T/M. \end{matrix} \square: \] We use \(K\) and \(F\) to denote the quotient fields on \(R\) and \(D\), respectively. The case \(k=F\) is of particular interest; in this case, we say that the diagram \(\square\) is of type \(\square^*\). The goal of this paper is to characterize certain coherentlike properties of integral domains in pullback constructions of type \(\square\). In one sense, this work is a sequel to that of J. W. Brewer and E. A. Rutter [Mich. Math. J. 23, 33-42 (1976; Zbl 0318.13007)], in which coherence, and several other properties are studied in so-called generalized \(D+M\) constructions.
An ideal \(I\) of a domain \(R\) is said to be \(v\)-finite if \(I^{-1}= J^{-1}\) for some finitely generated ideal \(J\) of \(R\). We denote section 2 to a study of divisoriality and \(v\)-finiteness in pullbacks of type \(\square\).
We then give a complete characterization of when \(M\) is \(v\)-finite as an ideal of \(R\).
A domain \(R\) is said to be \(v\)-coherent if \(I^{-1}\) is a \(v\)-finite divisorial ideal for each finitely generated ideal \(I\) of \(R\). Section 3 begins with a brief review of known facts, most of which are contained in the thesis of D. Nour el Abidine [“Groupe des classes de certains anneaux intègres et idéaux transformés” (Lyon 1992)]. We then proceed to characterize when \(R\) is \(v\)-coherent. In section 4, we use results and techniques from sections 2 and 3 to characterize several other coherentlike conditions in pullbacks of type \(\square\), e.g.:
\(R\) is coherent (quasicoherent, a finite conductor domain) if and only if exactly one of the following conditions holds:
(1) \(k\) is the quotient field of \(D\), \(D\) and \(T\) are coherent (quasicoherent, finite conductor domains), and \(T_M\) is a valuation domain; or
(2) \(D\) is a field, \([k:D] <\infty\), \(T\) is coherent (quasicoherent, a finite conductor domain), and \(M\) is finitely generated in \(T\).

MSC:

13B24 Going up; going down; going between (MSC2000)
13A15 Ideals and multiplicative ideal theory in commutative rings
13G05 Integral domains
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