相关论文: The reciprocal complements of classes of integral …

The *reciprocal complement* $R(D)$ of an integral domain $D$ is the subring of its fraction field generated by the reciprocals of its nonzero elements. Many properties of $R(D)$ are determined when $D$ is a polynomial ring in $n\geq 2$…

Given an integral domain $D$ with fraction field $F$, its *reciprocal complement* is the subring of $F$ generated by all $1/d$ for nonzero $d$ in $D$. This paper serves doubly as a survey of the current state of the field and an update with…

We study the reciprocal complement $\mathcal{R}(D)$ of a two-dimensional finitely generated $K$-algebra $D$ by linking it with the properties of a surface with coordinate ring $D$. We give several sufficient criteria to have…

We give a geometric interpretation of the reciprocal complement of an integral domain $D$ in the case $D$ is a one-dimensional finitely generated algebra over an algebraically closed field.

Let D be an integral domain with quotient field K. For any set X, the ring Int(D^X) of integer-valued polynomials on D^X is the set of all polynomials f in K[X] such that f(D^X) is a subset of D. Using the t-closure operation on fractional…

Let D be a Euclidean domain, with fraction field K. Let R(D) be the subring of K generated by the reciprocals of the nonzero elements of D. The main theorem states that if R(D) is not equal to K, then R(D) is a rank 1 discrete valuation…

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…

Let $D$ be an integral domain with quotient field $K,$ $E$ a subset of $K$ and $X$ an indeterminate over $K$. The set $\mathrm{Int}(E,D):=\{f\in K[X];\; f(E)\subseteq D\}$, of integer-valued polynomials on $E$ over $D$, is known to be an…

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…

Given an integral domain $D$ with quotient field $K$, the ring of integer-valued polynomials on D is the subring $\{f (X) \in K[X]: f(D) \subset D\}$ of the polynomial ring $K[X]$. Using the related tools of $t$-closure and associated…

Let D be a domain with quotient field K and A a D-algebra. We call a polynomial with coefficients in K that maps every element of A to an element of A "integer-valued on A". For commutative A we also consider integer-valued polynomials in…

Let $A\subset B$ be an integral ring extension of integral domains with fields of fractions $K$ and $L$, respectively. The integral degree of $A\subset B$, denoted by ${\rm d}_A(B)$, is defined as the supremum of the degrees of minimal…

Let $R$ be an integral domain with $qf(R)=K$ and let $F(R)$ be the set of nonzero fractional ideals of $R.$ Call $R$ a dually compact domain (DCD) if for each $I\in F(R)$ the ideal $I_{v}=(I^{-1})^{-1}$ is a finite intersection of principal…

We classify dp-minimal integral domains, building off the existing classification of dp-minimal fields and dp-minimal valuation rings. We show that if R is a dp-minimal integral domain, then R is a field or a valuation ring or arises from…

Let $D$ be an integral domain with quotient field $K$ and $E$ a subset of $K$. The \textit{ring of integer-valued rational functions on} $E$ is defined as $$\mathrm{int}_R(E,D):=\lbrace \varphi \in K(X);\; \varphi(E)\subseteq D\rbrace.$$…

In this paper, we completely describe the family of integrally closed Noetherian domains between $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$. We accomplish this result by classifying the Krull domains between these two polynomial rings. To this…

An ordered semiring is a commutative semiring equipped with a compatible preorder. Ordered semirings generalise both distributive lattices and commutative rings, and provide a convenient framework to unify certain aspects of lattice theory…

We study certain properties of modules over 1-dimensional local integral domains. First, we examine the order of the conductor ideal and its expected relationship with multiplicity. Next, we investigate the reflexivity of certain…

We show how to construct discretely ordered principal ideal subrings of $\mathbb Q[x]$ with various types of prime behaviour. Given any set $\mathcal D$ consisting of finite strictly increasing sequences $(d_1,d_2,\dots, d_l)$ of positive…

Let $F$ be a field, let $D$ be a subring of $F$ and let $Z$ be an irreducible subspace of the space of all valuation rings between $D$ and $F$ that have quotient field $F$. Then $Z$ is a locally ringed space whose ring of global sections is…