Narrow Results

In this paper, we introduce the notions of the directed, undirected, compressed directed, and compressed undirected irreducible divisor graphs $\mathrm{\Gamma }(x)$, $G(x)$, ${\mathrm{\Gamma }}_c(x),$ and $G_c(x)$ of a nonzero nonunit $x$ in a noncommutative atomic domain $D$, 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. Algebra35(3) (2007) 885895] and Axtell et al. [Irreducible divisor graphs and factorization properties of domains, Comm. Algebra39(11) (2011) 41484162] to the noncommutative setting.

A domain R is a maximal non-ACCP subring of its quotient field if and only if R is either a two-dimensional valuation domain with a DVR overring or a one-dimensional nondiscrete valuation domain. If R ⊂ S is a minimal ring extension and S is a domain, then (R,S) is a residually algebraic pair. If S is a domain but not a field, a maximal non-ACCP subring extension R ⊂ S is a minimal ring extension if (R,S) is a residually algebraic pair and R is quasilocal. Results with a similar flavor are given for domains R ⊂ S sharing a nonzero ideal, with applications to rings R of the form A + XB[X] or A + XB[[X]]. If R ⊂ S is a minimal ring extension such that R is a domain and S is not (R-algebra isomorphic to) an overring of R, then R satisfies ACCP if and only if S satisfies ACCP.

Let R ⊂ S be an extension of integral domains. We say that (R, S) is a well-centered pair of rings, if each intermediate ring T between R and S is well-centered on R, in the sense that each principal ideal of T is generated by an element of R. The aim of this paper is to study well-centered pairs of rings. We investigate the structure of the intermediate rings T between R and S that are well-centered on R. We establish the relationship between well-centered pairs and normal pairs. We present some examples and counterexamples illustrating our theory and showing the limits of our results.

Let $(R,M)$ be a quasilocal integral domain. We investigate the set of irreducible elements (atoms) of $R$. Special attention is given to the set of atoms in $M\setminus M^2$ and to the existence of atoms in $M^2$. While our main interest is in local Cohen–Kaplansky (CK) domains (atomic integral domains with only finitely many nonassociate atoms), we endeavor to obtain results in the greatest generality possible. In contradiction to a statement of Cohen and Kaplansky, we construct a local CK domain with precisely eight non-associate atoms having an atom in $M^2$.