Related papers: $\star $-super potent domains

An integral domain is called {\em Globalized multiplicatively pinched-Dedekind domain $($GMPD domain$)$} if every nonzero non-invertible ideal can be written as $JP_1\cdots P_k$ with $J$ invertible ideal and $P_1,...,P_k$ distinct ideals…

Let $f$ be a rational map with an infinitely-connected fixed parabolic Fatou domain $U$. We prove that there exists a rational map $g$ with a completely invariant parabolic Fatou domain $V$, such that $(f,U)$ and $(g,V)$ are conformally…

Let $\cal R$ be an ordered vector space over an ordered division ring. We prove that every definable set $X$ is a finite union of relatively open definable subsets which are definably simply-connected, settling a conjecture from [5]. The…

Let $(R, \mathfrak{m})$ be a regular local ring of dimension $d \geq 2$. A local monoidal transform of $R$ is a ring of the form $R_1= R[\frac{\mathfrak{p}}{x}]_{\mathfrak{m}_1}$ where $x \in \mathfrak{p}$ is a regular parameter,…

We study rings with infinitely (only finitely) many maximal subrings. We prove that if $M$ is a maximal left/right ideal of a ring $T$ which is not an ideal of $T$, and $R$ is the idealizer of $M$, then $T$ has at least $|R/M|+1$ maximal…

Using an extension of the abundancy index to imaginary quadratic rings that are unique factorization domains, we investigate what we call $n$-powerfully $t$-perfect numbers in these rings. This definition serves to extend the concept of…

Let T be a complete local ring and C a finite set of incomparable prime ideals of T. We find necessary and sufficient conditions for T to be the completion of an integral domain whose generic formal fiber is semilocal with maximal ideals…

Let L be the Leavitt path algebra of an arbitrary directed graph E over a field K. This survey article describes how this highly non-commutative ring L shares a number of the characterizing properties of a Dedekind domain or a Pr\"ufer…

It is a well-known and easily established fact that every Euclidean domain is also a principal ideal domain. However, the converse statement is not true, and this is usually shown by exhibiting as a counterexample the ring of algebraic…

Let $D$ be a Krull domain admitting a prime element with finite residue field and let $K$ be its quotient field. We show that for all positive integers $k$ and $1 < n_1 \leq \ldots \leq n_k$ there exists an integer-valued polynomial on $D$,…

We give conditions for a maximal divisorial ideal to be t-maximal and show with examples that, even in a completely integrally closed domain, maximal divisorial ideals need not be t-maximal.

Let $R$ be a maximal subring of a ring $T$, and $(R:T)$, $(R:T)_l$ and $(R:T)_r$ denote the greatest ideal, left ideal and right ideal of $T$ which are contained in $R$, respectively. It is shown that $(R:T)_l$ and $(R:T)_r$ are prime…

We study a condition on intersections of localizations of a domain at maximal t-ideals. This extends and generalizes earlier work of Gilmer (1967), Gilmer-Heinzer (1968), Olberding (1998), and others for Prufer domains.

A multiplicative subset $S$ of a ring $R$ is called \textit{strongly multiplicative} if $(\bigcap_{i\in\Delta}s_iR)\cap S \neq \emptyset$ for each family $(s_i)_{i\in\Delta}$ of elements in $S$. In this paper, we investigate how these sets…

Suppose that $R$ is a local domain essentially of finite type over a field of characteristic 0, and $\nu$ a valuation of the quotient field of $R$ which dominates $R$. The rank of such a valuation often increases upon extending the…

We define radically finite rings and show that finite dimensional radically finite rings are Noetherian, and that if either R is a finite character Hilbert domain that contains a field of characteristic zero or a finite dimensional Prufer…

We introduce and study the set of radical stable operations of an integral domain $D$. We show that their set is a complete lattice that is the join-completion of the set of spectral semistar operations, and we characterize when every…

This paper investigates t-reductions of ideals in pullback constructions. Section 2 examines the correlation between the notions of reduction and t-reduction in pseudo-valuation domains. Section 3 solves an open problem on whether the…

We construct a tree T of maximal degree 3 with infinitely many leaves such that whenever finitely many of them are removed, the remaining tree is isomorphic to T. In this sense T resembles an infinite star.

Let D be a Krull domain and Int(D) the ring of integer-valued polynomials on D. For any f in Int(D), we explicitly construct a divisor homomorphism from [f], the divisor-closed submonoid of Int(D) generated by f, to a finite sum of copies…