Related papers: Dp-minimal integral domains

Let A be an integral domain with field of fractions K. We investigate the structure of the overrings B of A (contained in K) that are well-centered on A in the sense that each principal ideal of B is generated by an element of A. We…

We study the topological spectrum of a seminormed ring $R$ which we define as the space of prime ideals $\mathfrak{p}$ such that $\mathfrak{p}$ equals the kernel of some bounded power-multiplicative seminorm. For any seminormed ring $R$ we…

In this paper, we advance an ideal-theoretic analogue of a "finite factorization domain" (FFD), giving such a domain the moniker "finite molecularization domain" (FMD). We characterize FMD's as those factorable domains (termed "molecular…

We prove that definable ring topologies on NIP fields are closely connected to NIP integral domains. More precisely, we show that up to elementary equivalence, any NIP topological field arises from an NIP integral domain. As an application,…

We generalize the usual relationship between irreducible Zariski closed subsets of the affine space, their defining ideals, coordinate rings, and function fields, to a non-commutative setting, where "varieties" carry a PGL_n-action, regular…

Let $R$ be a commutative $F$-algebra, where $F$ is a field of characteristic 0, satisfying the following conditions: $R$ is equidimensional of dimension $n$, every residual field with respect to a maximal ideal is an algebraic extension of…

A new class of rings, {\em the class of weakly left localizable rings}, is introduced. A ring $R$ is called {\em weakly left localizable} if each non-nilpotent element of $R$ is invertible in some left localization $S^{-1}R$ of the ring…

Let $R$ be a valuation ring and let $Q$ be its total quotient ring. It is proved that any singly projective (respectively flat) module is finitely projective if and only if $Q$ is maximal (respectively artinian). It is shown that each…

The main focus of this paper is on determining the highest non-vanishing local cohomology modules of \Omega_(B/R), \Omega_(B/V)(\Omega_(B/k)) where R is either a complete regular local ring or a complete local normal domain with coefficient…

The authors introduced the notion of $p_g$-ideals for two-dimensional excellent normal local domain over an algebraicaly closed field in terms of resolution of singularities. In this note, we give several ring-theoretic characterization of…

Let $V$ be a valuation ring of a global field $K$. We show that for all positive integers $k$ and $1 < n_1 \leq \ldots \leq n_k$ there exists an integer-valued polynomial on $V$, that is, an element of $\text{Int}(V) = \{ f \in K[X] \mid…

A minor is principal means it is defined by the same row and column indices. Let $X$ be a square generic matrix, $K[X]$ the polynomial ring in entries of $X$, over an algebraically closed field, $K$. For fixed $t\leq n$, let $\mathfrak P_t$…

In this paper, we introduce a new generalization of weakly prime ideals called $I$-prime. Suppose $R$ is a commutative ring with identity and $I$ a fixed ideal of $R$. A proper ideal $P$ of $R$ is $I$-prime if for $a, b \in R$ with $ab \in…

Let $D$ be a principal ideal domain with infinite spectrum such that for every nonzero prime ideal $M$ of $D$, the residue field $D/M$ is finite. Let $K$ be the quotient field of $D$. We investigate sets of lengths in the ring of…

We investigate a special class of Pr\"ufer domains, firstly introduced by Dress in 1965. The {\it minimal Dress ring} $D_K$, of a field $K$, is the smallest subring of $K$ that contains every element of the form $1/(1+x^2)$, with $x\in K$.…

Let $A$ be a ring and $R$ be a polynomial or a power series ring over $A$. When $A$ has dimension zero, we show that the Bass numbers and the associated primes of the local cohomology modules over $R$ are finite. Moreover, if $A$ has…

Given a square matrix $A$ with entries in a commutative ring $S$, the ideal of $S[X]$ consisting of polynomials $f$ with $f(A) =0$ is called the null ideal of $A$. Very little is known about null ideals of matrices over general commutative…

Let $(A,\mathfrak{m})$ be a complete Cohen-Macaulay local ring. Assume $A$ is not Gorenstein. We say $A$ is a Teter ring if there exists a complete Gorenstein ring $(B,\mathfrak{n})$ with $\dim B = \dim A$ and a surjective map $B…

For a commutative ring R we investigate the property that the sets of minimal primes of finitely generated ideals of R is always finite. We prove this property passes to polynomial ring extensions (in an arbitrary number of variables) over…

Let $(R,\m,k)$ be a local (Noetherian) ring of positive prime characteristic $p$ and dimension $d$. Let $G_\dt$ be a minimal resolution of the residue field $k$, and for each $i\ge 0$, let $\gothic t_i(R) = \lim_{e\to \8}…