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Let $H$ be a Krull monoid with class group $G$ such that every class contains a prime divisor. Then every nonunit $a \in H$ can be written as a finite product of irreducible elements. If $a=u_1 \cdot\ldots\cdot u_k$ with irreducibles $u_1, \ldots, u_k \in H$, then $k$ is called the length of the factorization and the set $\mathsf L (a)$ of all possible $k$ is the set of lengths of $a$. It is well-known that the system $\mathcal L (H) = \{\mathsf L (a) \mid a \in H \}$ depends only on the class group $G$. We study the inverse question asking whether the system $\mathcal L (H)$ is characteristic for the class group. Let $H'$ be a further Krull monoid with class group $G'$ such that every class contains a prime divisor and suppose that $\mathcal L (H) = \mathcal L (H')$. We show that, if one of the groups $G$ and $G'$ is finite and has rank at most two, then $G$ and $G'$ are isomorphic (apart from two well-known exceptions).
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참고문헌 (38)
참고문헌 신청
P. C. Baayen / 1969 / (C2 ⊕ C2 ⊕ C2 ⊕ C2n)! / Math. Centrum Amsterdam Afd. Zuivere Wisk 1969:21
N. R. Baeth / 2014 / Monoids of modules and arithmetic of direct-sum decompositions / Pacific J. Math 271(2):257~319
N. Baeth / 2009 / Atoms of the relative block monoid / Involve 2(1):29~36
N. R. Baeth / 2013 / Factorization theory and decompositions of modules / Amer. Math. Monthly 120(1):3~34
P. Baginski / 2013 / Products of two atoms in Krull monoids and arithmetical characterizations of class groups / European J. Combin 34(8):1244~1268
N. R. Baeth / 2014 / Monoids of modules and arithmetic of direct-sum decompositions / Pacific J. Math 271(2):257~319
N. Baeth / 2009 / Atoms of the relative block monoid / Involve 2(1):29~36
N. R. Baeth / 2013 / Factorization theory and decompositions of modules / Amer. Math. Monthly 120(1):3~34
P. Baginski / 2013 / Products of two atoms in Krull monoids and arithmetical characterizations of class groups / European J. Combin 34(8):1244~1268
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