Related papers: A Question about Differential Ideals

We consider the following conjecture: if X is a smooth projective variety over a field of characteristic zero, then there is a dense set of reductions X_s of X to positive characteristic such that the action of the Frobenius morphism on the…

We give examples of two dimensional normal ${\mathbb Q}$-Gorenstein graded domains, where the set of $F$-thresholds of the maximal ideal is not discrete, thus answering a question by Musta\c{t}\u{a}-Takagi-Watanabe. We also prove that, for…

In this note we give a negative answer to Abraham Robinson's question whether a finitely generated extension of an undecidable field is always undecidable. We construct 'natural' undecidable fields of transcendence degree 1 over Q all 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…

Suppose E/F is a field extension. We ask whether or not there exists an element of E whose characteristic polynomial has one or more zero coefficients in specified positions. We show that the answer is frequently ``no''. We also prove…

In this paper, our purpose is to define the expansion of $r$-hyperideals and extend this concept to $\phi$-$\delta$-$r$-hyperideal. Let $\Re$ be a commutative Krasner hyperring with nonzero identity. Given an expansion $\delta$ of…

Transfer Krull monoids are a recent concept including all commutative Krull domains and also, for example, wide classes of non-commutative Dedekind domains. We show that transfer Krull monoids are fully elastic (i.e., every rational number…

In this paper, for a field $k$ of characteristic zero and a finitely generated $k$-algebra $R$, we give a set of generators for the image ideals of irreducible nice and quasi-nice $R$-derivations on the polynomial ring $R[X,Y]$, where $R$…

For an algebraic function field $F/K$ and a discrete valuation $v$ of $K$ with perfect residue field $k$, we bound the number of discrete valuations on $F$ extending $v$ whose residue fields are algebraic function fields of genus zero over…

Let $\ast $ be a finite character star operation defined on an integral domain $D.$ Call a nonzero $\ast $-ideal $I$ of finite type a $\ast $ -homogeneous ($\ast $-homog) ideal, if $I\subsetneq D$ and $(J+K)^{\ast }\neq D$ for every pair…

The main aim of this paper is to characterize ideals I in the power series ring R=K[[x1,...,xs]] that are finitely determined up to contact equivalence by proving that this is the case if and only if I is an isolated complete intersection…

Normal ideals on regular uncountable cardinals are familiar objects. We investigate ideals that are pleasant--while a normal ideal is closed under arbitrary diagonal unions, a pleasant ideal is closed only under diagonal unions indexed by…

Let R be a commutative Noetherian local ring, and denote by mod R the category of finitely generated R-modules. In this paper, we consider when mod R has a nontrivial extension-closed subcategory. We prove that this is the case if there are…

Let $R$ be a commutative ring with ${\Bbb{A}}(R)$ its set of ideals with nonzero annihilator. In this paper and its sequel, we introduce and investigate the {\it annihilating-ideal graph} of $R$, denoted by ${\Bbb{AG}}(R)$. It is the…

Let $R$ be a commutative ring, $Y\subseteq \mathrm{Spec}(R)$ and $ h_Y(S)=\{P\in Y:S\subseteq P \}$, for every $S\subseteq R$. An ideal $I$ is said to be an $\mathcal{H}_Y$-ideal whenever it follows from $h_Y(a)\subseteq h_Y(b)$ and $a\in…

Let R be a commutative ring. If P is a maximal ideal of R whose a power is finitely generated then we prove that P is finitely generated if R is either locally coherent or arithmetical or a polynomial ring over a ring of global dimension…

There has arisen in recent years a substantial theory of "multiplier ideals'' in commutative rings. These are integrally closed ideals with properties that lend themselves to highly interesting applications. But how special are they among…

We show that there exist noncommutative Ore extensions in which every right ideal is two-sided. This answers a problem posed by Marks in Duo Rings and Ore extensions, J.Algebra 280(2), (2004). We also provide an easy construction of one…

Let R be a commutative ring with identity and I be an ideal of R. The cozero-divisor graph with respect to I, denoted by $\Gamma''_I(R)$, is the graph of R with vertices {x \in R -I :xR +I \not=R} and two distinct vertices $x$ and $y$ are…

We study a class of first-order theories whose complete quantifier-free types with one free variable either have a trivial positive part or are isolated by a positive quantifier-free formula--plus a few other technical requirements. The…