Related papers: Semistar Dedekind Domains
We study stable semistar operations defined over a Pr\"ufer domain, showing that, if every ideal of a Pr\"ufer domain $R$ has only finitely many minimal primes, every such closure can be described through semistar operations defined on…
We consider the lattice-ordered groups Inv$(R)$ and Div$(R)$ of invertible and divisorial fractional ideals of a completely integrally closed Pr\"ufer domain. We prove that Div$(R)$ is the completion of the group Inv$(R)$, and we show there…
Let $D$ be an integral domain and $\star $ a star operation defined on $D$. We say that $D$ is a $\star $-power conductor domain ($\star $-PCD) if for each pair $a,b\in D\backslash (0)$ and for each positive integer $n$ we have $Da^{n}\cap…
In this article, we study the monoid of fractional ideals and the ideal class semigroup of an arbitrary given one dimensional normal domain O obtained by an infinite integral extension of a Dedekind domain. We introduce a notion of "upper…
Let $D$ be an integral domain and $\star$ a semistar operation stable and of finite type on it. In this paper, we are concerned with the study of the semistar (Krull) dimension theory of polynomial rings over $D$. We introduce and…
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…
We study different form of boundness for ideals of almost Dedekind domains, generalizing the notions of critical ideals, radical factorization, and SP-domains. We show that every almost Dedekind domain has at least one noncritical maximal…
Call a semistar operation $\ast$ on the polynomial domain $D[X]$ an extension (respectively, a strict extension) of a semistar operation $\star$ defined on an integral domain $D$, with quotient field $K$, if $E^\star = (E[X])^{\ast}\cap K$…
Let $R$ be a normal Noetherian local domain of Krull dimension two. We examine intersections of rank one discrete valuation rings that birationally dominate $R$. We restrict to the class of prime divisors that dominate $R$ and show that if…
We study almost Dedekind domains with respect to the failure of ideals to have radical factorization, that is, we study how to measure how far an almost Dedekind domain is from being an SP-domain. To do so, we consider the maximal space…
We generalize the theory of radical factorization from almost Dedekind domain to strongly discrete Pr\"ufer domains; we show that, for a fixed subset $X$ of maximal ideals, the finitely generated ideals with $\mathcal{V}(I)\subseteq X$ have…
We introduce and study the notion of $\star$-stability with respect to a semistar operation $\star$ defined on a domain $R$; in particular we consider the case where $\star$ is the $w$-operation. This notion allows us to generalize and…
We provide a complete classification of all tilting modules and tilting classes over almost perfect domains, which generalizes the classifications of tilting modules and tilting classes over Dedekind and 1-Gorenstein domains. Assuming the…
It is well-known that a ring is Noetherian if and only if every ascending chain of ideals is stationary, and an integral domain is a PID if and only if every countably generated ideal is principal. We respectively investigate the similar…
It is well-known that an integrally closed domain $D$ can be express as the intersection of its valuation overrings but, if $D$ is not a Pr\"{u}fer domain, the most of valuation overrings of $D$ cannot be seen as localizations of $D$. The…
Let $R$ be an integral domain, $Star(R)$ the set of all star operations on $R$ and $StarFC(R)$ the set of all star operations of finite type on $R$. Then $R$ is said to be star regular if $|Star(T)|\leq |Star(R)|$ for every overring $T$ of…
In 1994, Matsuda and Okabe introduced the notion of semistar operation, extending the "classical" concept of star operation. In this paper, we introduce and study the notions of semistar linkedness and semistar flatness which are natural…
Let $R$ be a commutative integral domain and let $\star$ be a semistar operation of finite type on $R$, and $I$ be a quasi-$\star$-ideal of $R$. We show that, if every minimal prime ideal of $I$ is the radical of a $\star$-finite ideal,…
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 I be a nonzero proper ideal in a Noetherian integral domain R. In this paper we establish the existence of a finite separable integral extension domain A of R and a positive integer m such that all the Rees integers of IA are equal to…