Journal of Algebraic Combinatorics

, Volume 37, Issue 3, pp 523–543

Zero-divisor graphs of nilpotent-free semigroups

Article

Abstract

We find strong relationships between the zero-divisor graphs of apparently disparate kinds of nilpotent-free semigroups by introducing the notion of an Armendariz map between such semigroups, which preserves many graph-theoretic invariants. We use it to give relationships between the zero-divisor graph of a ring, a polynomial ring, and the annihilating-ideal graph. Then we give relationships between the zero-divisor graphs of certain topological spaces (so-called pearled spaces), prime spectra, maximal spectra, tensor-product semigroups, and the semigroup of ideals under addition, obtaining surprisingly strong structure theorems relating ring-theoretic and topological properties to graph-theoretic invariants of the corresponding graphs.

Keywords

Zero-divisor graph Armendariz map Graph invariants Annihilating-ideal graph Comaximal graph Nilpotent-free semigroup 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Institut für MathematikUniversität OsnabrückOsnabrückGermany
  2. 2.Department of Engineering Science, Faculty of EngineeringUniversity of TehranTehranIran

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