The integral-closedness criterion for integer-valued polynomial rings on algebras

Let DD be an ADDB-domain. Let AA be a DD-algebra that is torsion-free and finitely generated as a DD-module, with AK=DA \cap K = D, and let B=ADKB=A\otimes_D K. Thus DD and AA satisfy the hypotheses of the main theorem.

Integral-closedness criterion. If IntK(A)\operatorname{Int}_K(A) is integrally closed, then IntK(A)\operatorname{Int}_K(A) is Prüfer.

The theorem preceding this conjecture gives several equivalent conditions for IntK(A)\operatorname{Int}_K(A) to be Prüfer, including its equality with IntK(A,A)\operatorname{Int}_K(A,A') and the condition A=AA=A'. The conjecture asks whether integral closedness alone is sufficient in the general ADDB-domain setting; the full resolution remains open.

Sources & referencesView supporting material

Primary source

Giulio Peruginelli and Nicholas J. Werner, “A classification of Prufer domains of integer-valued polynomials on algebras”, arXiv:2509.09243 (2026).

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