Types of irreducible divisor graphs of noncommutative domains

In this paper, we introduce the notions of the directed, undirected, compressed directed, and compressed undirected irreducible divisor graphs [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] of a nonzero nonunit [Formula: see text] in a noncommutative atomic domain [Formula: see text], respectively. In light of these notions, we identify the corresponding characterizations of the normal unique factorization domain (n-UFD). Moreover, we study the connection between the kinds of directed and undirected irreducible divisor graphs and an n-FFD. Consequently, these characterizations generalize the results of Coykendall and Maney [Irreducible divisor graphs, Comm. Algebra 35(3) (2007) 885895] and Axtell et al. [Irreducible divisor graphs and factorization properties of domains, Comm. Algebra 39(11) (2011) 41484162] to the noncommutative setting.