Some results on integer-valued polynomials over modules
Bull. Korean Math. Soc.
Published online July 9, 2020
Ali Reza Naghipour and Javad Sedighi Hafshejani
Shahrekord University
Abstract : 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 $\Int(R,M)\neq M[X]$. Among the other results, it is shown that $\Int(\mathbb{Z},M)$ is not Noetherian module over $\Int_{M}(\mathbb{Z})\cap\Int(\mathbb{Z})$, where $M$ is a finitely generated $\mathbb{Z}$-module.
Keywords : Integer-valued polynomial, Noetherian ring, Krull dimension, polynomially torsion-free.
MSC numbers : 11C99, 13A99, 13B25, 13F20.
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