Narrow Results

Call a domain R an sQQR-domain if each simple overring of R, i.e., each ring of the form R[u] with u in the quotient field of R, is an intersection of localizations of R. We characterize Prüfer domains as integrally closed sQQR-domains. In the presence of certain finiteness conditions, we show that the sQQR-property is very strong; for instance, a Mori sQQR-domain must be a Dedekind domain. We also show how to construct sQQR-domains which have (non-simple) overrings which are not intersections of localizations.

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.

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.

An integral domain R with field of fractions K is called a maximal non-1-catenarian subring of K if the polynomial ring in one variable, R[X] is not catenarian and for each proper intermediate ring T (that is each ring T such that R ⊂ T ⊆ K) T[X] is catenarian. The main purpose of this paper is to prove that the concept of maximal non-1-catenarian subrings and that of maximal non-universally catenarian subrings are equivalent.

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.

Let R ⊆ T be a (unital) extension of (commutative) rings, such that the total quotient ring of R is a von Neumann regular ring and T is torsion-free as an R-module. Let T ⊆ B be a ring extension such that B is a reduced ring that is torsion-free as a T-module. Let R* (respectively, A) be the integral closure of R in T (respectively, in B). Then (R*, T) is a normal pair (i.e. S is integrally closed in T for each ring S such that R* ⊆ S ⊆ T) if and only if (A, AT) is a normal pair. This generalizes results of Prüfer and Heinzer on Prüfer domains to normal pairs of complemented rings.

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.

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.

In this paper, we characterize maximal non-ACCP subrings R of a domain S in case (R, S) is a residually algebraic pair and R is semilocal. In this paper, we also consider a K-algebra S, a nonzero proper ideal I of S and a subring D of the field K and we determine necessary and sufficient conditions in order that D + I is a maximal non-ACCP subring of S. This gives an example of a maximal non-ACCP subring R of a domain S such that (R, S) is a normal pair and R is not semilocal.

Let $R$ be an integral domain with only finitely many overrings, equivalently, a domain such that its integral closure $R^{\prime }$ is a Prüfer domain with finite spectrum and there are only finitely many rings between $R$ and $R^{\prime }$. Jaballah solved the problem of counting the overrings in the case $R=R^{\prime }$ but left the general case as an open problem [A. Jaballah, The number of overrings of an integrally closed domain, Expo. Math.23 (2005) 353–360, Problem 3.4]. The purpose of this paper is to provide a solution to that problem.

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.

If $A\subseteq B$ are (commutative) rings, $[A,B]$ denotes the set of intermediate rings and $(A,B)$ is called an almost valuation (AV)-ring pair if each element of $[A,B]$ is an AV-ring. Many results on AV-domains and their pairs are generalized to the ring-theoretic setting. Let $R\subseteq S$ be rings, with ${\overline{R}}_S$ denoting the integral closure of $R$ in $S$. Then $(R,S)$ is an AV-ring pair if and only if both $(R,{\overline{R}}_S)$ and $({\overline{R}}_S,S)$ are AV-ring pairs. Characterizations are given for the AV-ring pairs arising from integrally closed (respectively, integral; respectively, minimal) ring extensions $R\subseteq S$. If $(R,S)$ is an AV-ring pair, then $R\subseteq S$ is a P-extension. The AV-ring pairs $(R,S)$ arising from root extensions are studied extensively. Transfer results for the “AV-ring” property are obtained for pullbacks of $(B,I,D)$ type, with applications to pseudo-valuation domains, integral minimal ring extensions, and integrally closed maximal non-AV subrings. Several sufficient conditions are given for $(R,S)$ being an AV-ring pair to entail that $S$ is an overring of $R$, but there exist domain-theoretic counter-examples to such a conclusion in general. If $(R,S)$ is an AV-ring pair and $R\subseteq S$ satisfies FCP, then each intermediate ring either contains or is contained in ${\overline{R}}_S$. While all AV-rings are quasi-local going-down rings, examples in positive characteristic show that an AV-domain need not be a divided domain or a universally going-down domain.

The main purpose of this paper is to answer a question which was left open in N. Jarboui and S. Aljubran [Maximal non-integrally closed subrings of an integral domain, Ric. Mat. (2020), doi.org/10.1007/s11587-020-00500-0] asking for a characterization of maximal non-integrally closed subrings of arbitrary rings.