Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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On the Relationship between Zero-sums and Zero-divisors of Semirings

  • Hetzel, Andrew J. (Department of Mathematics, Tennessee Technological University) ;
  • Lufi, Rebeca V. Lewis (Department of Mathematics, Tennessee Technological University)
  • Received : 2007.07.09
  • Accepted : 2008.07.27
  • Published : 2009.06.30
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Abstract

In this article, we generalize a well-known result of Hebisch and Weinert that states that a finite semidomain is either zerosumfree or a ring. Specifically, we show that the class of commutative semirings S such that S has nonzero characteristic and every zero-divisor of S is nilpotent can be partitioned into zerosumfree semirings and rings. In addition, we demonstrate that if S is a finite commutative semiring such that the set of zero-divisors of S forms a subtractive ideal of S, then either every zero-sum of S is nilpotent or S must be a ring. An example is given to establish the existence of semirings in this latter category with both nontrivial zero-sums and zero-divisors that are not nilpotent.

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Cited by

  1. Morita invariants of semirings related to a Morita context pp.1793-7183, 2019, https://doi.org/10.1142/S1793557119500232