Ideals In Almost Distributive Lattice

Abstract

This thesis aims to develop a better understanding of Almost distributive lattice and its ideals. We present the definition of Almost distributive lattice, ideals and filters. Also we study some basic properties of Almost distributive lattice and give some examples on this class which includes almost all the existing ring theoretic generalisations of a Boolean ring like regular rings. The concepts of a-ideals, annihilator ideals, O-ideals and minimal prime ideals are defined, and we furnish the relation between these concepts. In addition It is proved that the set of all annihilator ideals of an Almost distributive lattice forms a complete Boolean algebra and the set of all a-ideals forms a complete distributive lattice. Also characterization theorems for each of these concepts are proved. Furthermore we present the definition of regular and 1r-regular ring, which are type of rings included by ADLs, The properties and characterizations theorems connecting between these concepts are studied, and several examples are given