Local tameness of v-noetherian monoids

Abstract Let H be a v -noetherian monoid, e.g., the multiplicative monoid R ∖ { 0 } of a noetherian domain R . We show that, for every b ∈ H , there exists a constant ω ( H , b ) ∈ N 0 having the following property: If n ∈ N and a 1 , … , a n ∈ H such that b divides the product a 1 ⋅ … ⋅ a n , then b already divides a subproduct of a 1 ⋅ … ⋅ a n consisting of at most ω ( H , b )  factors. Using the ω ( H , ⋅ ) -quantities we derive a new characterization of local tameness–a crucial finiteness property in the theory of non-unique factorizations.

[1]  A. Geroldinger,et al.  Arithmetic of Mori domains and monoids , 2008 .

[2]  A. Geroldinger.  Chains of factorizations in weakly Krull domains , 1997 .

[3]  Alfred Geroldinger,et al.  Non-unique factorizations , 2006 .

[4]  Luca Diracca.  On a generalization of the exchange property to modules with semilocal endomorphism rings , 2007 .

[5]  Wolfgang Hassler,et al.  On the Arithmetic of Strongly Primary Monoids , 2007 .

[6]  S. Chapman,et al.  Half-Factorial Domains, a Survey , 2000 .

[7]  J. Huckaba.  Commutative Rings with Zero Divisors , 1988 .

[8]  Alfred Geroldinger,et al.  Non-Unique Factorizations : Algebraic, Combinatorial and Analytic Theory , 2006 .

[9]  Florian Kainrath.  On Local Half-Factorial Orders , 2005 .

[10]  W. Hassler,et al.  Factorization of powers in C-monoids , 2006 .

[11]  S. Chapman,et al.  On the arithmetic of arithmetical congruence monoids , 2007 .

[12]  D. Gronau,et al.  Half-factorial sets in finite abelian groups : a survey , 2006 .

[13]  László Fuchs,et al.  Infinite Abelian groups , 1970 .

[14]  Franz Halter-Koch,et al.  Ideal Systems: An Introduction to Multiplicative Ideal Theory , 1998 .

[15]  D. Michel,et al.  Répartition des idéaux premiers parmi les classes d'idéaux dans un anneau de Dedekind et équidécomposition , 1986 .

[16]  Joaquín Pascual,et al.  Infinite Abelian Groups , 1970 .

[17]  S. Chapman,et al.  The catenary and tame degree of numerical monoids , 2009 .

[18]  J. Coykendall.  Extensions of Half-Factorial Domains: A Survey , 2005 .

[19]  L. Skula.  HALF-FACTORIAL SUBSETS IN INFINITE ABELIAN GROUPS , 2003 .

[20]  Alfred Geroldinger,et al.  Monotone Chains of Factorizations in C-Monoids , 2005 .

[21]  J. Brewer,et al.  Multiplicative Ideal Theory in Commutative Algebra , 2006 .

[22]  P. A. Grillet.  Commutative Nilsemigroups with Zero Cohomology , 2001 .

[23]  W. Hassler,et al.  Big indecomposable modules and direct-sum relations , 2007 .

[24]  W. Hassler,et al.  Direct-sum decompositions over one-dimensional Cohen-Macaulay local rings , 2006 .

[25]  S. Yau.  Mathematics and its applications , 2002 .

[26]  Weidong Gao,et al.  Zero-sum problems in finite abelian groups: A survey , 2006 .