Narrow Results

This paper presents a complete characterization of an edge-weighted simple graph $G_{\omega }$, whose edge ideal $I(G_{\omega })$ is integrally closed. Furthermore, it is demonstrated that if $I(G_{\omega })$ is integrally closed and $G_{\omega }$ is an edge-weighted star graph, path, or cycle, then $I(G_{\omega })$ is normal.

In this paper, we deal with the study of maximal non-universally catenarian subrings (that is a non-universally catenarian subring R of a domain S such that any subring of S that properly contains R is universally catenarian). Several satisfactory results are given, and for S a field and R not integrally closed, we characterize these domains.

Results of Davis on normal pairs (R, T) of domains are generalized to (commutative) rings with nontrivial zero-divisors, particularly complemented rings. For instance, if T is a ring extension of an almost quasilocal complemented ring R, then (R, T) is a normal pair if and only if there is a prime ideal P of R such that T = R_[P], R/P is a valuation domain and PT = P. Examples include sufficient conditions for the "normal pair" property to be stable under formation of infinite products and ⋈ constructions.

We establish several characterizations of maximal non-Prüfer and maximal non-integrally closed subrings of a field. Special attention is given when finiteness conditions are satisfied. Several numerical characterizations are then obtained. The paper is concluded with examples that exhibit the obtained results.

As an extension of the class of Dedekind domains, we have introduced and studied the class of multiplicatively pinched-Dedekind domains (MPD domains) and the class of Globalized multiplicatively pinched-Dedekind domains (GMPD domains) ([T. Dumitrescu and S. U. Rahman, A class of pinched domains, Bull. Math. Soc. Sci. Math. Roumanie52 (2009) 41–55] and [T. Dumitrescu and S. U. Rahman, A class of pinched domains II, Comm. Algebra39 (2011) 1394–1403]). The main interest of this paper is to study GMPD domains that have only finitely many overrings.

Let $R\subseteq S$ be an extension of integral domains with only finitely many intermediate rings, where $R$ is not a field and $S$ is not necessarily the quotient field of $R$ or $R$ is not necessarily integrally closed in $S$. In this paper, we exactly determine the number of intermediate rings between $R$ and $S$ and give a way to compute it.

Let $R\subset S$ be an extension of integral domains. The ring $R$ is said to be maximal non-Prüfer subring of $S$ if $R$ is not a Prüfer domain, while each subring of $S$ properly containing $R$ is a Prüfer domain. Jaballah has characterized this kind of ring extensions in case $S$ is a field [A. Jaballah, Maximal non-Prüfer and maximal non-integrally closed subrings of a field, J. Algebra Appl.11(5) (2012) 1250041, 18 pp.]. The aim of this paper is to deal with the case where $S$ is any integral domain which is not necessarily a field. Several examples are provided to illustrate our theory.