Types of Irreducible Divisor Graphs of Noncommutative Domains, II

Abstract

In this paper, we continue investigation of the directed and undirected irreducible divisor graph concepts $G(x)$ and $\Gamma (x)$ of $x\in D^{\ast }\backslash U(D)$, respectively, which were introduced in [7]. Consequently, we introduce two generalizations of these concepts. The first one is the irreducible divisor simplicial complex $S(x)$ of $x\in D^{\ast }\backslash U(D)$ in a noncommutative atomic domain $D$, which simultaneously extends the commutative case that was introduced by R. Baeth and J. Hobson in [3]. The second one is the directed and undirected $\tau$-irreducible divisor graphs $G_{\tau }(x)$ and ${\Gamma }_{\tau }(x)$ of $x\in D^{\ast }\backslash U(D)$, respectively, in a noncommutative $\tau$-atomic domain $D$ with a symmetric and associate preserving relation $\tau$ on $D^{\ast }\backslash U(D)$. Those graphs also extend the commutative case that was introduced by C. P. Mooney in [5]. Furthermore, we extend the results of [3] and [5] to give a characterization of n-unique factorization domains via those two generalizations.