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On the Relationship between Zero-sums and Zero-divisors of Semirings
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  • Journal title : Kyungpook mathematical journal
  • Volume 49, Issue 2,  2009, pp.221-233
  • Publisher : Department of Mathematics, Kyungpook National University
  • DOI : 10.5666/KMJ.2009.49.2.221
 Title & Authors
On the Relationship between Zero-sums and Zero-divisors of Semirings
Hetzel, Andrew J.; Lufi, Rebeca V. Lewis;
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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.
semiring;idealization;nilpotent element;primal ideal;primary ideal;prime ideal;subtractive ideal;zero-divisor;zero-sum;zerosumfree;
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