Related papers: An introduction to reciprocal complements of integ…
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 quotient field $\mathcal{Q}(D)$, the reciprocal complement of $D$ is the subring $R(D)$ of $\mathcal{Q}(D)$ whose elements are all the sums $\frac{1}{d_1}+\ldots+\frac{1}{d_n} $ for $d_1, \ldots, d_n$…
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.
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…
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…
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 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…
In this paper, we exhibit the creation of the maximal integral domain mid(R) generated by a nonzero commutative ring R.
Inverse semigroups are a class of semigroups whose structure induces a compatible partial order. This partial order is examined so as to establish mirror properties between an inverse semigroup and the semilattice of its idempotent…
We study those integral domains in which every proper ideal can be written as an invertible ideal multiplied by a nonempty product of proper radical ideals.
A generating function for reciprocal binomial coefficients is written down, integral representations of this function are obtained, generating functions for sums of reciprocal binomial coefficients are derived, new identities are obtained,…
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…
The concept of integral as an inverse to that of derivation was already introduced for rings and recently also for lattices. Since semirings generalize both rings and bounded distributive lattices, it is natural to investigate integration…
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…
This note intended to give a counterexample to a question related to the following theorem. Let D be a differential domain finitely generated over a differential field F with algebraically closed field of constants,C, of characteristic 0.…
Let $S\subseteq R$ be a multiplicatively closed subset of a ring $R$. We extend several results on integral domains to their $S$-versions and establish the $S$-version of Krull intersection theorem. We also show that if $R$ is an $S$-field,…
We prove a continued fraction expansion for the reciprocal of a certain $q$-series. All the specialists in the world are asked whether it is new or not.
This paper proves a reciprocity formula for modular inverses for non-zero integers and demonstrates some applications of the reciprocity formula in calculating or verifying some modular inverses of specific forms, including the modular…
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 describe the additive subgroups of fields which are closed with respect to taking inverses. In particular, in characteristic different from two any such subgroup is either a subfield or the kernel of the trace map of a quadratic…