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Let $R$ be a factorial domain. In this work we investigate the connections between the arithmetic of ${\rm Int}(R)$ (i.e., the ring of integer-valued polynomials over $R$) and its monadic submonoids (i.e., monoids of the form $\{g\in {\rm Int}(R)\mid g\mid_{{\rm Int}(R)} f^k$ for some $k\in\mathbb{N}_0\}$ for some nonzero $f\in {\rm Int}(R)$). Since every monadic submonoid of ${\rm Int}(R)$ 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 ${\rm Int}(R)$ 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 ${\rm Int}(R)$ can be explained by using the structure of monadic submonoids of ${\rm Int}(R)$.
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참고문헌 (8)
참고문헌 신청
A. Geroldinger / 2006 / Non-unique factorizations: Algebraic, combinatorial and analytic theory / Chapman and Hall/CRC : xxii+700 ~
A. Reinhart / 2014 / Com-mutative algebra / Springer : 307 ~ 330
A. Reinhart / 2016 / A note on conductor ideals / Comm. Algebra 44 (10) : 4243 ~ 4251
F. Kainrath / 1999 / Factorization in Krull monoids with infinite class group / Colloq. Math 80 (1) : 23 ~ 30
P.-J. Cahen / 1997 / Integer-valued polynomials / American Mathematical Society
A. Reinhart / 2014 / Com-mutative algebra / Springer : 307 ~ 330
A. Reinhart / 2016 / A note on conductor ideals / Comm. Algebra 44 (10) : 4243 ~ 4251
F. Kainrath / 1999 / Factorization in Krull monoids with infinite class group / Colloq. Math 80 (1) : 23 ~ 30
P.-J. Cahen / 1997 / Integer-valued polynomials / American Mathematical Society
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