相关论文: $t$-local domains and valuation domains

The notion of maximal non valuative domain is introduced and characterized. An integral domain R is called a maximal non valuative domain if R is not a valuative domain but every proper overring of R is a valuative domain. Maximal non…

It is shown that every dp-minimal integral domain $R$ is a local ring and for every non-maximal prime ideal $\mathfrak p $ of $R$, the localization $R_{\mathfrak p }$ is a valuation ring and $\mathfrak{p}R_{\mathfrak{p}}=\mathfrak{p}$.…

Let $A$ be an integral domain. We study new conditions on families of integral ideals of $A$ in order to get that $A$ is of $t$-finite character (i.e., each nonzero element of $A$ is contained in finitely many $t$-maximal ideals). We also…

If $R$ is a valuation domain of maximal ideal $P$ with a maximal immediate extension of finite rank it is proven that there exists a finite sequence of prime ideals $P=L_0\supset L_1\supset...\supset L_m\supseteq 0$ such that…

It is shown that a local ring R of bounded module type is an almost maximal valuation ring if there exists a non-maximal prime ideal J such that R/J is a maximal valuation domain.

It is well-known that if R is a domain with finite character, each locally principal nonzero ideal of R is invertible. We address the problem of understanding when the converse is true and survey some recent results.

It is proved that the localization of an injective module E, over a valuation ring R, at a prime ideal J, is injective if J is not the subset of zero-divisors of R or if J or E is flat. It follows that localizations of injective modules…

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.

We show that in certain Pr\"ufer domains, each nonzero ideal $I$ can be factored as $I=I^v \Pi$, where $I^v$ is the divisorial closure of $I$ and $\Pi$ is a product of maximal ideals. This is always possible when the Pr\"ufer domain is…

It is proved that if $R$ is a valuation domain with maximal ideal $P$ and if $R_L$ is countably generated for each prime ideal $L$, then $R^R$ is separable if and only $R_J$ is maximal, where $J=\cap_{n\in\mathbb{N}}P^n$.

Let $T$ be a complete local (Noetherian) ring of characteristic zero. We find necessary and sufficient conditions for $T$ to be the completion of a quasi-excellent local domain. In the case that $T$ contains the rationals, we provide…

We show that if $R$ is a local domain which is dominated by a valuation $\nu$, then there does not always exist a regular local ring $R'$ which birationally dominates R and is dominated by $\nu$ and an extension of $\nu$ to the…

Let $\Gamma$ be a torsionless commutative cancellative monoid, $R =\bigoplus_{\alpha \in \Gamma}R_{\alpha}$ be a $\Gamma$-graded integral domain, and $H$ be the set of nonzero homogeneous elements of $R$. In this paper, we show that if $Q$…

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.

For a finite-type star operation $\star$ on a domain $R$, we say that $R$ is $\star$-super potent if each maximal $\star$-ideal of $R$ contains a finitely generated ideal $I$ such that (1) $I$ is contained in no other maximal $\star$-ideal…

An integral domain $D,$ with quotient field $K,$ is a $v$-domain if for each nonzero finitely generated ideal $A$ of $D$ we have $(AA^{-1})^{-1}=D.$ It is well known that if $D$ is a $v$-domain$,$ then some quotient ring $D_{S}$ of $D$ may…

We classify dp-minimal integral domains, building off the existing classification of dp-minimal fields and dp-minimal valuation rings. We show that if R is a dp-minimal integral domain, then R is a field or a valuation ring or arises from…

In this article we revisit a problem regarding Bezout domains, namely, whether every Bezout domain is an elementary divisor domain. We prove that a Bezout domain in which every maximal ideal is principal is an elementary divisor ring

An integral domain $D$ is a {\em valuation ideal factorization domain} (VIFD) if each nonzero principal ideal of $D$ can be written as a finite product of valuation ideals. Clearly, $\pi$-domains are VIFDs. We study the ring-theoretic…

A commutative integral domain is primary if and only if it is one-dimensional and local. A domain is strongly primary if and only if it is local and each nonzero principal ideal contains a power of the maximal ideal. Hence one-dimensional…