• Bulletin of the Korean Mathematical Society
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• v.39 no.1
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• pp.165-174
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• 2002

A polynomial if k-fold integer-valued if itself and its derivatives up to order k are integer-valued. Necessary and sufficient conditions are established far (i) polynomials with rational coefficients to be k-fold integer-valued, and for (ii) a quotient of two k-fold integer-valued polynomials to be k-fold integer-valued.

• Bulletin of the Korean Mathematical Society
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• v.57 no.5
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• pp.1165-1176
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• 2020

Let M be a module over a commutative ring R. In this paper, we study Int(R, M), the module of integer-valued polynomials on M over R, and Int_M(R), the ring of integer-valued polynomials on R over M. We establish some properties of Krull dimensions of Int(R, M) and Int_M(R). We also determine when Int(R, M) and Int_M(R) are nontrivial. Among the other results, it is shown that Int(ℤ, M) is not Noetherian module over Int_M(ℤ) ∩ Int(ℤ), where M is a finitely generated ℤ-module.

• Journal of the Korean Mathematical Society
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• v.38 no.5
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• pp.915-935
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• 2001

Let V be any valuation domain and let E be a subset of the quotient field K of V. We study the ring of integer-valued polynomials on E, that is, Int(E, V)={f$\in$K[X]｜f(E)⊆V}. We show that, if E is precompact, then Int(E, V) has many properties similar to those of the classical ring Int(Z).In particular, Int(E, V) is dense in the ring of continuous functions C(E, V); each finitely generated ideal of Int(E, V) may be generated by two elements; and finally, Int(E, V) is a Prufer domain.

• Communications of the Korean Mathematical Society
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• v.32 no.2
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• pp.233-260
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• 2017

Let R be a factorial domain. In this work we investigate the connections between the arithmetic of Int(R) (i.e., the ring of integer-valued polynomials over R) and its monadic submonoids (i.e., monoids of the form {$g{\in}Int(R){\mid}g{\mid}_{Int(R)}f^k$ for some $k{\in}{\mathbb{N}}_0$} for some nonzero $f{\in}Int(R)$). Since every monadic submonoid of Int(R) is a Krull monoid it is possible to describe the arithmetic of these monoids in terms of their divisor-class group. We give an explicit description of these divisor-class groups in several situations and provide a few techniques that can be used to determine them. As an application we show that there are strong connections between Int(R) and its monadic submonoids. If $R={\mathbb{Z}}$ or more generally if R has sufficiently many "nice" atoms, then we prove that the infinitude of the elasticity and the tame degree of Int(R) can be explained by using the structure of monadic submonoids of Int(R).