## 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

### Citations:

Zbl 0324.13001; Zbl 0318.13007
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