Related papers: Domains whose ideals meet a universal restriction

Let $D$ be an integral domain. Then $D$ is an almost valuation (AV-)domain if for $a, b\in D\setminus \{0\}$ there exists a natural number $n$ with $a^{n}\mid b^{n}$ or $b^{n}\mid a^{n}$. AV-domains are closely related to valuation domains,…

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…

Using the idea of quasi-ideals of $P$-regular nearrings, the concept of bi-ideals of $P$-regular nearrings is generalized, which is an extension of the concept of quasi-ideals of $P$-regular nearrings and some interesting characterizations…

A ring R is said to be of stable range 1.5 if for each a, b from R and nonzero c from R satisfying aR + bR + cR = R there exists r from R such that (a + br)R + cR = R. Let R be a commutative domain in which all finitely generated ideals are…

In this paper, we prove that a domain which verifies some integral inequality is either (strictly) contained in the solution of some free boundary problem, or it coincides with an $N$-ball. We also present new overdetermined value problems…

The inaccessibility of a point p in a bounded domain D \subset R^n is the minimum of the lengths of segments through p with boundary at \bd D. The points of maximum inaccessibility I_D are those where the inaccessibility achieves its…

Let $\mathcal{I}$ be an analytic P-ideal [respectively, a summable ideal] on the positive integers and let $(x_n)$ be a sequence taking values in a metric space $X$. First, it is shown that the set of ideal limit points of $(x_n)$ is an…

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…

Let $(A,{\mathfrak m})$ be a Cohen-Macaulay local ring and let $I$ be an ideal of $A$. We prove that the Rees algebra ${\mathcal R}(I)$ is an almost Gorenstein ring in the following cases: (1) $(A,{\mathfrak m})$ is a two-dimensional…

We expand the notion of core to $cl$-core for Nakayama closures $cl$. In the characteristic $p>0$ setting, when $cl$ is the tight closure, denoted by *, we give some examples of ideals when the core and the *-core differ. We note that…

Let $A$ be a residually finite dimensional algebra (not necessarily associative) over a field $k$. Suppose first that $k$ is algebraically closed. We show that if $A$ satisfies a homogeneous almost identity $Q$, then $A$ has an ideal of…

We investigate the higher divisorial ideal $D(I):= Ann(Ext^g_R(R/I,R))$ associated to an ideal I of grade g. Our main focus is the containment problem $D(I) \subseteq \overline{I}$. We show that this inclusion holds for broad classes of…

When $I$ is the radical homogeneous ideal of a finite set of points in projective $N$-space, ${\bf P}^N$, over a field $K$, it has been conjectured that $I^{(rN-N+1)}$ should be contained in $I^r$ for all $r\geq 1$. Recent counterexamples…

An infinite integral domain $R$ is called a large ideal domain (LID) if every nontrivial ideal of $R$ has finite index in $R$. Recently, N. Hindman and D. Strauss have established a refinement of Moreira's theorem for the set of natural…

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

Let $X$ be an integral scheme of finite type over a complete DVR of mixed characteristic. We provide a definition of a test ideal which agrees with the multiplier ideal after inverting $p$, is computed from a sufficiently large alteration,…

Let $R=\oplus_{m\geq 0}R_m$ be a standard graded equidimensional ring over a field $R_0$, and $I\subseteq J$ be two non-nilpotent graded ideals in $R$. Then we give a set of numerical characterizations of the integral dependence of $I$ and…

We introduce and explore the Uniform Izumi-Rees Property in Noetherian rings with applications to multiplicity theory and containment relationships among symbolic powers of ideals. As an application, we prove that if $R$ is a normal domain…

In a Dedekind domain $D$, every non-zero proper ideal $A$ factors as a product $A=P_1^{t_1}\cdots P_k^{t_k}$ of powers of distinct prime ideals $P_i$. For a Dedekind domain $D$, the $D$-modules $D/P_i^{t_i}$ are uniserial. We extend this…

In this study, we study the null controllability of a multi-dimensional degenerate parabolic equation characterized by a degenerate interior point. The control domain, which is an arbitrary inner region, does not encompass the degenerate…