Related papers: Presentations and module bases of integer-valued p…

Let $D$ be an integrally closed domain with quotient field $K$ and $n$ a positive integer. We give a characterization of the polynomials in $K[X]$ which are integer-valued over the set of matrices $M_n(D)$ in terms of their divided…

Let $K$ be a number field of degree $n$ with ring of integers $O_K$. By means of a criterion of Gilmer for polynomially dense subsets of the ring of integers of a number field, we show that, if $h\in K[X]$ maps every element of $O_K$ 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…

Given a star operation * of finite type, we call a domain R a *-unique representation domain (*-URD) if each *-invertible *-ideal of R can be uniquely expressed as a *-product of pairwise *-comaximal ideals with prime radical. When * is the…

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 $R$ be either the ring of Lipschitz quaternions, or the ring of Hurwitz quaternions. Then, $R$ is a subring of the division ring $\mathbb{D}$ of rational quaternions. For $S \subseteq R$, we study the collection $\rm{Int}(S,R) = \{f \in…

Let $I \subset R = \mathbb{F}[x_1,x_2]$ be a height two ideal minimally generated by three homogeneous polynomials of the same degree $d$, where $\mathbb{F}$ is a field of characteristic zero. We use the theory of $D$-modules to deduce…

We develop a new technique for studying monomial ideals in the standard polynomial rings $A[X_1,\ldots,X_d]$ where $A$ is a commutative ring with identity. The main idea is to consider induced ideals in the semigroup ring…

This is an expository paper in which it is proved that, for every infinite field ${\mathbf{F}}$, the polynomial ring ${\mathbf{F}}[t_1,\ldots, t_n]$ has Krull dimension $n$. The proof uses only "high school algebra" and the rudiments of…

We show that every polynomial overring of the ring ${\rm Int}(\mathbb Z)$ of polynomials which are integer-valued over $\mathbb Z$ may be considered as the ring of polynomials which are integer-valued over some subset of $\hat{\mathbb{Z}}$,…

Let $S$ be a subset of $\overline{\mathbb Z}$, the ring of all algebraic integers. A polynomial $f \in \mathbb Q[X]$ is said to be integral-valued on $S$ if $f(s) \in \overline{\mathbb Z}$ for all $s \in S$. The set $\text{Int}_{\mathbb…

Using polynomial evaluation, we give some useful criteria to answer questions about divisibility of polynomials. This allows us to develop interesting results concerning the prime elements in the domain of coefficients. In particular, it is…

An integral domain $D$ is a $v$--domain if, for every finitely generated nonzero (fractional) ideal $F$ of $D$, we have $(FF^{-1})^{-1}=D$. The $v$--domains generalize Pr\"{u}fer and Krull domains and have appeared in the literature with…

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 $\mathfrak P_t$…

To study the question of whether every two-dimensional Pr\"ufer domain possesses the stacked bases property, we consider the particular case of the Pr\"ufer domains formed by integer-valued polynomials. The description of the spectrum of…

We study the algebraic and arithmetic structure of monoids of invertible ideals (more precisely, of $r$-invertible $r$-ideals for certain ideal systems $r$) of Krull and weakly Krull Mori domains. We also investigate monoids of all nonzero…

Let $A$ and $B$ be commutative rings with identity. An {\it $A$-$B$-biring} is an $A$-algebra $S$ together with a lift of the functor $Hom_A(S,-)$ from $A$-algebras to sets to a functor from $A$-algebras to $B$-algebras. An {\it…

We review the definition of D-rings introduced by H. Gunji & D. L. MacQuillan. We provide an alternative characterization for such rings that allows us to give an elementary proof of that a ring of algebraic integers is a D-ring. Moreover,…

Let $F$ be an algebraically closed field of positive characteristic and let $R$ be a finitely generated $F$-algebra with a filtration with the property that the associated graded ring of $R$ is an integral domain of Krull dimension two. We…

Let $R$ be an integral domain. For elements $a,b \in R$, let $[a,b]$ denote their greatest common divisor, if it exists. We say that $R$ has the Z-property if whenever $a,b,c,d$ and $e$ are nonzero nonunits of $R$ such that $abc=de$, then…