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
