2011/06/02 by Tongsuo Wu, Wu, Tongsuo, Dancheng Lu +3
Mathematics · #Advanced Topics in Algebra #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1106.0348
openalex publication_date 2011/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. A po-semiring is a semiring equipped with a compatible bounded partial order. In this paper, properties of zero divisors and prime elements of a po-semiring are studied. In particular, it is proved that under some mild assumption the set Z(A) of nonzero zero divisors of A is A∖ \0,1\, each prime element of A is a maximal element, and the zero divisor graph \G(A) of A is a finite graph if and only if A is finite. For a po-semiring A with Z(A)=A∖ \0,1\, it is proved that A has finitely many maximal elements if ACC holds either for elements of A or for principal annihilating ideals of A. As applications of prime elements, it is shown that the structure of a po-semiring A is completely determined by the structure of integral po-semirings if either |Z(A)|=1 or |Z(A)|=2 and Z(A)2\not=0. Applications to the ideal structure of commutative rings are considered.