A BIPARTITE GRAPH ASSOCIATED WITH IRREDUCIBLE ELEMENTS AND GROUP OF UNITS IN Z_n

Abstract

A nonzero nonunit a of a ring R is called an irreducible element if, for some b, c ∈ R, a = bc implies that either b or c (not both) is a unit. We construct a bipartite graph in which the union of the set of irreducible elements and group of units is a vertex-set and an edge-set is the set of pairs between irreducible elements and their unit factors in the ring of integers modulo n. Many properties of this constructed bipartite graph are studied. We show that this bipartite graph contains components which are isomorphic. We also note that each component of this bipartite graph can be presented in some form which we call star form presentation. Some examples of graphs in star form presentation are provided for illustration purposes. Furthermore, we prove that the girth of this bipartite graph is 8. Most of the results in this paper are arrived at via group action.