Distributive lattices with strong endomorphism kernel property as direct sums

Saeid Bagheri ,Mahtab Koohi Kerahroodi

doi:10.29252/cgasa.14.1.223

Abstract

In this paper,for a distributive lattice $mathcal L$, we study and compare some lattice theoretic features of $mathcal L$ and topological properties of the Stone spaces ${rm Spec}(mathcal L)$ and ${rm Max}(mathcal L)$ with the corresponding graph theoretical aspects of the zero-divisor graph $Gamma(mathcal L)$.Among other things,we show that the Goldie dimension of $mathcal L$ is equal to the cellularity of the topological space ${rm Spec}(mathcal L)$ which is also equal to the clique number of the zero-divisor graph $Gamma(mathcal L)$. Moreover, the domination number of $Gamma(mathcal L)$ will be compared with the density and the weight of the topological space ${rm Spec}(mathcal L)$. For a $0$-distributive lattice $mathcal L$, we investigate the compressed subgraph $Gamma_E(mathcal L)$ of the zero-divisor graph $Gamma(mathcal L)$ and determine some properties of this subgraph in terms of some lattice theoretic objects such as associated prime ideals of $mathcal L$.

Highlights

In 1988, Beck [6] introduced the zero-divisor graph Γ0(R) of a commutative ring R whose vertices are elements of R and two distinct vertices x and y are adjacent if and only if xy = 0
The main purpose of this process was to obtain a subgraph of the zero-divisor graph of a ring which preserves many properties of the original graph, but it is easier to deal with because it has smaller vertex and edge sets
At first for a distributive lattice L, we study and investigate some relations among lattice theoretic featurs of L such as Goldie dimension and minimal prime ideals of L and some topological properties of the Stone spaces Spec(L) and Max(L) like cellularity, density and weight of them and compare them with some graph theoretic aspects of the zerodivisor graph Γ(L) such as its clique number and domination number

Summary

Introduction

In Theorem 3.1, we establish equalities among the Goldie dimension of a distributive lattice L, the cellularity of its Stone space Spec(L) and the clique number of the corresponding zero-divisor graph. We state more results on the zero-divisor graph Γ(L) of a distributive lattice L relating it with topological properties of Spec(L) and Max(L) and some lattice theoretic aspects of L.

Results

Conclusion