Related papers: Radical factorization in higher dimension

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…

Let $S \subset R$ be an arbitrary subset of a unique factorization domain $R$ and $\K$ be the field of fractions of $R$. The ring of integer-valued polynomials over $S$ is the set $\mathrm{Int}(S,R)= \{ f \in \mathbb{K}[x]: f(a) \in R\…

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…

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…

An integral domain $D$ is called an SP-domain if every ideal is a product of radical ideals. Such domains are always almost Dedekind domains, but not every almost Dedekind domain is an SP-domain. The SP-rank of $D$ provides a natural…

Analytic properties of function spaces over the real and the complex fields are different in some ways. This reflects in algebraic properties which are different at times and similar in some other respects. For instance, the ring of…

In recent years, centrally essential rings have been intensively studied in ring theory. In particular, they find applications in homological algebra, group rings, and the structural theory of rings. The class of essentially central rings…

Let T be a complete local (Noetherian) equidimensional ring with maximal ideal m such that the Krull dimension of T is at least two and the depth of T is at least two. Suppose that no integer of T is a zerodivisor and that |T|=|T/m|. Let d…

Ideals in Leavitt path algebras have been shown to share many properties with those of integral domains. Since studying factorizations of ideals in integral domains into special types of ideals (particularly, prime, prime-power, primary,…

In this paper, we study the classes of rings in which every proper (regular) ideal can be factored as an invertible ideal times a nonempty product of proper radical ideals. More precisely, we investigate the stability of these properties…

Let $(R,\mathfrak{m})$ be a local Noetherian ring with residue field $k$. While much is known about the generating sets of reductions of ideals of $R$ if $k$ is infinite, the case in which $k$ is finite is less well understood. We…

Fix any field $K$ of characteristic $p$ such that $[K:K^p]$ is finite. We discuss excellence for Noetherian domains whose fraction field is $K$, showing for example, that $R$ is excellent if and only if the Frobenius map is finite on $R$.…

Given a Noetherian ring $A$, the collection of all integrally closed ideals in $A$ which contain a nonzerodivisor, denoted $ic(A)$, forms a cancellative monoid under the operation $I*J=\overline{IJ}$, the integral closure of the product.…

For a given family $(G_i)_{i \in \N}$ of finitely generated abelian groups, we construct a Dedekind domain $D$ having the following properties. \begin{enumerate} \item $\Pic(D) \cong \bigoplus_{i \in \N}G_i$. \item For each $i \in \N$,…

The ring of periodic distributions on ${\mathbb{R}}^{\tt d}$ with usual addition and with convolution is considered. Via Fourier series expansions, this ring is isomorphic to the ring ${\mathcal{S}}'({\mathbb{Z}}^{\tt d})$ of all maps…

We prove in this article the surjectivity of three maps. We prove in Theorem $1.6$ the surjectivity of the Chinese remainder reduction map associated to the projective space of an ideal with a given factorization into ideals whose radicals…

Let $S$ be a commutative noetherian ring. The extensions of matrix factorizations of non-zerodivisors $x_1,\dots,x_n$ of $S$ form a full subcategory of finitely generated modules over the quotient ring $S/(x_1\cdots x_n)$. In this paper, we…

In this paper, we obtain some factorization results on formal power series over principle ideal domains with sharp bounds on number of irreducible factors. These factorization results correspondingly lead to irreducibility criteria for…

Pr\"{u}fer domains and subclasses of integral domains such as Dedekind domains admit characterizations by means of the properties of their ideal lattices. Interestingly, a Leavitt path algebra $L$, in spite of being non-commutative and…

The primary purpose of this paper is give a classification scheme for the nonzero primes of a Pr\"ufer domain based on five properties. A prime $P$ of a Pr\"ufer domain $R$ could be sharp or not sharp, antesharp or not, divisorial or not,…