A Survey on Flatness in Integer-Valued Polynomial Rings

Ali Tamoussit ()
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Ali Tamoussit: The Regional Center for Education and Training Professions Souss-Massa, Department of Mathematics

A chapter in Algebraic, Number Theoretic, and Topological Aspects of Ring Theory, 2023, pp 443-461 from Springer

Abstract: Abstract For an integral domain D with quotient field K, the ring of integer-valued polynomials over D, denoted by Int(D), consists of all polynomials f in K[X] such that f(D) ⊆ D. This type of rings has attracted significant attention. In particular, Cahen et al. in [Open Problems in Commutative Ring Theory, in: Commutative Algebra: Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions, M. Fontana, S. Frisch and S. Glaz (editors), Springer (2014), pp. 293–305] asked whether Int(D) is always flat as a D-module. Yet, this natural question remains unanswered. In this survey article, we collect some old and recent results on (faithfull) flatness of integer-valued polynomial rings, and we give some illustrating examples.

Date: 2023
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DOI: 10.1007/978-3-031-28847-0_23

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