[Paper Review] On zero divisors and prime elements of po-semirings

Tongsuo Wu, Dancheng Lu+1 more|arXiv (Cornell University)|Jun 2, 2011

Rings, Modules, and Algebras10 references3 citations

TL;DR

This paper investigates zero divisors and prime elements in bounded semirings, proving that under mild conditions, the set of nonzero zero divisors equals $A \setminus \{0,1\}$, and each prime element is maximal. It establishes that if $Z(A) = A \setminus \{0,1\}$, then $A$ has finitely many maximal elements under ACC on elements or principal annihilating ideals, and completely characterizes bounded semirings with $|Z(A)| = 1$ or $|Z(A)| = 2$ and $Z(A)^2 \neq 0$ via integral 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 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 $A\setminus \{0,1\}$, 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 $Z(A)=A\setminus \{0,1\}$, 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 $Z(A)^2 ot=0$. Applications to the ideal structure of commutative rings are considered.

Motivation & Objective

• To analyze the structure of zero divisors and prime elements in bounded semirings.
• To determine conditions under which the set of nonzero zero divisors equals $A \setminus \{0,1\}$.
• To prove that each prime element in a bounded semiring is maximal under mild assumptions.
• To characterize bounded semirings with $|Z(A)| = 1$ or $|Z(A)| = 2$ and $Z(A)^2 \neq 0$ in terms of integral bounded semirings.
• To explore applications to the ideal structure of commutative rings via bounded semiring theory.

Proposed method

• Uses the poset structure of bounded semirings, leveraging their local semimodularity and shellability of the chain complex.
• Applies the notion of bounded semirings as dioids with $a + a = a$, where the partial order is defined by $a \leq b \iff a + b = b$.
• Introduces and analyzes prime elements via the condition: $xy \leq p$ implies $x \leq p$ or $y \leq p$, with $p \neq 1$.
• Employs the concept of lower principal ideals $<u> = \{x \in A \mid x \leq u\}$ and annihilating ideals $\text{ann}_A(u) = \{x \in A \mid xu = 0\}$.
• Applies condition $(C_3)$: for non-nilpotent $u$, there exists a nonzero idempotent $w \leq u$ with an orthogonal idempotent complement.
• Constructs explicit models of bounded semirings with $|Z(A)| = 2$ and $Z(A)^2 = 0$ using integral bounded semirings $A_1$ and additional elements $c, u$.

Experimental results

Research questions

• RQ1 Under what conditions is the set of nonzero zero divisors of a bounded semiring equal to $A \setminus \{0,1\}$?
• RQ2 When is every prime element in a bounded semiring maximal?
• RQ3 What is the structure of a bounded semiring when $|Z(A)| = 1$ or $|Z(A)| = 2$ and $Z(A)^2 \neq 0$?
• RQ4 How does the ACC on elements or principal annihilating ideals affect the finiteness of maximal elements in a bounded semiring with $Z(A) = A \setminus \{0,1\}$?
• RQ5 Can the ideal structure of commutative rings be recovered or analyzed through the framework of bounded semirings?

Key findings

• The set of nonzero zero divisors $Z(A)$ equals $A \setminus \{0,1\}$ under mild assumptions, implying that all elements other than 0 and 1 are zero divisors.
• Each prime element in a bounded semiring is maximal, meaning no proper element lies strictly between a prime and 1.
• If $Z(A) = A \setminus \{0,1\}$, then $A$ has finitely many maximal elements when either ACC holds on elements or on principal annihilating ideals.
• A bounded semiring with $|Z(A)| = 1$ or $|Z(A)| = 2$ and $Z(A)^2 \neq 0$ is completely determined by the structure of an integral bounded semiring.
• The paper constructs explicit models of bounded semirings with $|Z(A)| = 2$ and $Z(A)^2 = 0$ using a base integral bounded semiring $A_1$ and two new elements $c, u$ with specific addition and multiplication rules.
• Under DCC and condition $(C_3)$, a bounded semiring $A$ is isomorphic to either $A_1$ or $\{0,1\}^{(n)} \times A_1$, where $A_1$ satisfies $c^2 = 0$ for all minimal elements $c$ and DCC holds on $A_1$.