Related papers: Valuation Ideal Factorization Domains
Let $D$ be a Pr\"ufer $\star$-multiplication domain, where $\star$ is a semistar operation on $D$. We show that certain ideal-theoretic properties related to idempotence and divisoriality hold in Pr\"ufer domains, and we use the associated…
This paper studies the Ratliff-Rush closure of ideals in integral domains. By definition, the Ratliff-Rush closure of an ideal $I$ of a domain $R$ is the ideal given by $\tilde{I}:=\bigcup(I^{n+1}:_{R}I^{n})$ and an ideal $I$ is said to be…
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 paper, we study conditions guaranteeing that a product of ideals defines a Golod ring. We show that for a $3$-dimensional regular local ring (or $3$-variable polynomial ring) $(R , \m)$, the ideal $I \m$ always defines a Golod ring…
A study of the relation between a noetherian local domain with a given valuation and its associated graded ring with respect to the valuation, which in some cases is an esentially toric variety, possibly of infinite embedding dimension, but…
A finite semifield $D$ is a finite nonassociative ring with identity such that the set $D^*=D\setminus\{0\}$ is closed under the product. In this paper we obtain a computer-assisted description of all 64-element finite semifields, which…
We discuss the Japanese and universally Japanese properties for valuation rings and Pr\"ufer domains. These properties, regarding finiteness of integral closure, have been studied extensively for Noetherian rings, but very rarely, if ever,…
In this paper we exhibit an example of a three-dimensional regular local domain (A, n) having a height-two prime ideal P with the property that the extension PA^ of P to the n-adic completion A^ of A is not integrally closed. We use a…
We study functions from a unique factorization monoid to a field. The set of all such functions is a commutative ring isomorphic to a ring of formal power series over the field, with indeterminates indexed by the prime elements of the…
In this paper we study the concept of radical factorization in the context of abstract ideal theory in order to obtain a unified approach to the theory of factorization into radical ideals and elements in the literature of commutative…
Let $T$ be a complete local (Noetherian) ring. For each $i \in \mathbb{N}$, let $C_i$ be a nonempty countable set of nonmaximal pairwise incomparable prime ideals of $T$, and suppose that if $i \neq j$, then either $C_i = C_j$ or no element…
It is well known that a domain without proper strongly divisorial ideals is completely integrally closed. In this paper we show that a domain without {\em prime} strongly divisorial ideals is not necessarily completely integrally closed,…
Finite dynamical systems (FDSs) are commonly used to model systems with a finite number of states that evolve deterministically and at discrete time steps. Considered up to isomorphism, those correspond to functional graphs. As such, FDSs…
Let $R$ be a commutative ring with identity. The small finitistic dimension $\fPD(R)$ of $R$ is defined to be the supremum of projective dimensions of $R$-modules with finite projective resolutions. In this paper, we characterize a ring $R$…
The study of prime divisibility plays a crucial role in number theory. The $p$-adic valuation of a number is the highest power of a prime, $p$, that divides that number. Using this valuation, we construct $p$-adic valuation trees to…
Let (R; m; k) be a local noetherian domain with field of fractions K and R_v a valuation ring, dominating R (not necessarily birationally). Let v|K be the restriction of v to K; by definition, v|K is centered at R. Let \hat{R} denote the…
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…
We investigate the inner vertex-isoperimetric problem on the $d$-regular tree $T_d$. We first determine the exact value of the inner vertex-isoperimetric profile $I_d(k) = \min\{ |\partial D| \mid D\subset T_d \text{ finite and connected},\…
We call a right module $M$ (strongly) virtually regular if every (finitely generated) cyclic submodule is isomorphic to a direct summand. $M$ is said to be completely virtually regular if every submodule is virtually regular. In this paper,…
The behavior of factorization properties in various ring extensions is a central theme in commutative algebra. Classically, the UFDs are (completely) integrally closed and tend to behave well in standard ring extensions, with the notable…