ON THE GENUS OF THE INTERSECTION GRAPH OF IDEALS OF A COMMUTATIVE RING

To each commutative ring R one can associate the graph G(R), called the intersection graph of ideals, whose vertices are nontrivial ideals of R. In this paper, we try to establish some connections between commutative ring theory and graph theory, by study of the genus of the intersection graph of ideals. We classify all graphs of genus 2 that are intersection graphs of ideals of some commutative rings and obtain some lower bounds for the genus of the intersection graph of ideals of a nonlocal commutative ring.

ON THE NOTION OF STRONG IRREDUCIBILITY AND ITS DUAL

This note gives a unifying characterization and exposition of strongly irreducible elements and their duals in lattices. The interest in the study of strong irreducibility stems from commutative ring theory, while the dual concept of strong irreducibility had been used to define Zariski-like topologies on specific lattices of submodules of a given module over an associative ring. Based on our lattice theoretical approach, we give a unifying treatment of strong irreducibility, dualize results on strongly irreducible submodules, examine its behavior under central localization and apply our theory to the frame of hereditary torsion theories.