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An inequality concerning the elasticity of Krull monoids with divisor class group $\mathbb{Z}_p$

Authors:

Institution:

(1) Trinity University, Department of Mathematics, One Trinity Place, San Antonio, Texas 78212-7200,;(2) The University of North Carolina at Chapel Hill, Department of Mathematics, Chapel Hill, North Carolina 27599-3250,

Abstract:

Abstract Let p be a prime integer and M a Krull monoid with divisor class group $\mathbb{Z}_p$ . We represent by S the set of nontrivial divisor classes of $\mathbb{Z}_p$ which contain prime divisors. We present a new inequality for the elasticity of M (denoted ρ (M)) which is dependent on the cardinality of S and argue that this inequality is the best possible. If M as above has | S| = 3, then it is known that $\rho(M)\in \{p/2, p/3,\ldots ,p/(\lfloor p/3\rfloor + 1)\}$ , but for large p, not all the values in this containment set can be realized. For each | S| = 3, we produce a submonoid $\widetilde{\mathcal{B}}(\mathbb{Z}_p,S)$ of $\mathcal{B}(\mathbb{Z}_p,S)$ such that

$\{\,\rho(\widetilde{\mathcal{B}}(\mathbb{Z}_p,S))\,\mid \, S\subseteq \mathbb{Z}_p-\{0\}\mbox{ and }\mid S\mid =3\}=\{p/2, p/3,\ldots ,p/(\lfloor p/3\rfloor + 1)\}.$

Keywords: Krull monoid, Block monoid, Elasticity of factorization Mathematics Subject Classification (2000): 20M14, 20D60, 13F05

Keywords:

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