On the Relationship between Zero-sums and Zero-divisors of Semirings

Abstract

Abstract. In this article, we generalize a well-known result of Hebisch and Weinert thatstates that a finite semidomain is either zerosumfree or a ring. Specifically, we show thatthe class of commutative semirings S such that S has nonzero characteristic and everyzero-divisor of S is nilpotent can be partitioned into zerosumfree semirings and rings. Inaddition, we demonstrate that if S is a finite commutative semiring such that the set ofzero-divisors of S forms a subtractive ideal of S, then either every zero-sum of S is nilpo-tent or S must be a ring. An example is given to establish the existence of semirings in thislatter category with both nontrivial zero-sums and zero-divisors that are not nilpotent. 1. IntroductionThis article is devoted to an exploration of how ideal-theoretic considerations incommutative semirings, particularly finite commutative semirings, impact the mul-tiplicative behavior of those elements of the semiring that have additive inversesin the semiring. The general question as to the algebraic nature of these so-called“zero-sums” of a semiring is one of the most central in the theory of semirings. Weare especially motivated by a result of Hebisch and Weinert [9, Corollary 3.4, p. 81]that establishes that the class of finite semidomains can be partitioned by the an-tipodal properties of being zerosumfree (that is, only the zero element is a zero-sumof the semiring) and being a ring (where, by definition, every element is a zero-sumof the semiring). Of course, there exist infinite semidomains with nontrivial zero-sums that are not rings; for example, the polynomial semiring XZ[X] + N, where