Zero divisors and prime elements of bounded semirings

Abstract

A semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. A bounded semiring is a semiring equipped with a compatible bounded partial order. In this paper, properties of zero divisors and prime elements of a bounded 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}, and each prime element of A is a maximal element. For a bounded 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 an application of prime elements, we show that the structure of a bounded semiring A is completely determined by the structure of integral bounded semirings if either |Z(A)| = 1 or |Z(A)| = 2 and Z(A) 2 = 0. Applications to the ideal structure of commutative rings are also considered. In particular, when R has a finite number of ideals, it is shown that the chain complex of the poset I(R) is pure and shellable, where I(R) consists of all ideals of R.

Commutative Rings R Whose C(𝔸𝔾(R)) Consist of Only Triangles

“…Notice that each minimal element of A is a zero-divisor if there exist more than one minimal element. Refer to [14] for more details about bounded semirings.…”

“…In Section 2, we recall the concept of bounded semirings as [14] and then give a quick proof showing that the zero-divisor graph A of a bounded semiring A is in if and only if A sits in the following set…”

The Structure of Finite Local Principal Ideal Rings

“…Note that our structure theorems as well as the proofs are quite different from [5]. The present work is motivated by works of [7,8].…”