Quotient Field Research Articles

On rings of integer-valued rational functions

Let D ⊆ B be an extension of integral domains and E a subset of the quotient field of D. We introduce the ring of D -valued B-rational functions on E, denoted by Int B R ( E , D ) , which naturally extends the concept of integer-valued polynomials, defined as Int B R ( E , D ) : = { f ∈ B ( X ) ; f ( E ) ⊆ D } . The notion of Int B R ( E , D ) boils down to the usual notion of integer-valued rational functions when the subset E is infinite. In this paper, we aim to investigate various properties of these rings, such as prime ideals, localization, and the module structure. Furthermore, we study the transfer of some ring-theoretic properties from Int R ( E , D ) to D.

Relative perinormality and relative global perinormality in a new pullback construction and idealization

If (S, J) is a zero-dimensional local ring, K = S / J its residue class field and D an integral domain with quotient field K, we show that twin notions of relative perinormality (N. Epstein, J. Shapiro, Perinormality in pullbacks Journal of Commutative Algebra 11 (3):341–362, 2019) and relative global perinormality (H. Klawa, Globally perinormal domains and graded perinormality, Doctoral Dissertation, George Mason University, 2023) are preserved by a pullback construction R = D × K S . As a byproduct, we recover the preservation of these notions in a new pullback construction of Chang et al. and idealization.

Lengths of factorizations of integer-valued polynomials on Krull domains with prime elements

Let D be a Krull domain admitting a prime element with finite residue field and let K be its quotient field. We show that for all positive integers k and 1<n1≤⋯≤nk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1 < n_1 \\le \\cdots \\le n_k$$\\end{document}, there exists an integer-valued polynomial on D, that is, an element of Int(D)={f∈K[X]∣f(D)⊆D}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{Int}\\,}}(D) = \\{ f \\in K[X] \\mid f(D) \\subseteq D \\}$$\\end{document}, which has precisely k essentially different factorizations into irreducible elements of Int(D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{Int}\\,}}(D)$$\\end{document} whose lengths are exactly n1,…,nk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n_1, \\ldots , n_k$$\\end{document}. Using this, we characterize lengths of factorizations when D is a unique factorization domain and therefore also in case D is a discrete valuation domain. This solves an open problem proposed by Cahen, Fontana, Frisch, and Glaz.

Analysis of carbamate and organophosphorus pesticide in agricultural irrigation water by Liquid Chromatography - Mass Spectrometry

So far, the ultra-trace concentration of pesticide residues in environmental samples has challenged many analytical techniques, which are used to detect simultaneously organophosphorus and carbamate with various polarities. In this study, the OASIS-HLB column was modified by the reagent 1,8-Dihydroxy-2-(4-sulfophenylazo)naphthalene-3,6-disulfonic to enhance the retention of polar pesticides. Both real irrigation water and distilled water samples were applied to validate the determination method of ten carbamates and thirty-eight organophosphorus pesticides all together using LC-MS. According to the results, the linear interval of the method ranged from 1.0 to 100 µg/L, the detection limit was as low as 5.0 µg/L, the relative standard deviations presented less than 8.0%, and the recoveries were ranging between 60% and 112%. Moreover, in the irrigation water samples which were collected in both dry and rainy seasons in ten stations around the agricultural area, five compounds were found, including fenamiphos, terbufos, aldicarb, propoxur, and methiocarb. In the midst of those sampling areas, the Environmental Impact Quotient Field Use Rating of detected pesticide residues presented a high value - 272 at one station, which indicates a high risk for the surrounding environment as well as the local people’s health.

Regular t-ideals of polynomial rings and semigroup rings with zero divisors

For an integrally closed domain D with quotient field K, it is well known that every fractional t-ideal of D[X] has the form hI[X] with h∈K(X) and I a t-ideal of D. We generalize this useful result to certain polynomial rings and semigroup rings with zero divisors. As an application of our work, we characterize the semigroup rings with torsion-free cancellative monoids of exponents that are Prüfer v-multiplication rings or (generalized) greatest common divisor rings. By showing that Glaz's (generalized) greatest common divisor ring property ascends to polynomial rings, we affirmatively settle one of her conjectures.

Quasi-inner automorphisms of Drinfeld modular groups

Let A be the set of elements in an algebraic function field K over \mathbb{F}_{q} which are integral outside a fixed place \infty . Let G=\mathrm{GL}_{2}(A) be a Drinfeld modular group . The normalizer of G in \mathrm{GL}_{2}(K) , where K is the quotient field of A , gives rise to automorphisms of G , which we refer to as quasi-inner . Modulo the inner automorphisms of G , they form a group \mathrm{Quinn}(G) which is isomorphic to \mathrm{Cl}(A)_{2} , the 2 -torsion in the ideal class group \mathrm{Cl}(A) . The group \mathrm{Quinn}(G) acts on all kinds of objects associated with G . For example, it acts freely on the cusps and elliptic points of G . If \mathcal{T} is the associated Bruhat–Tits tree, the elements of \mathrm{Quinn}(G) induce non-trivial automorphisms of the quotient graph G \backslash \mathcal{T} , generalizing an earlier result of Serre. It is known that the ends of G \backslash \mathcal{T} are in one-to-one correspondence with the cusps of G . Consequently, \mathrm{Quinn}(G) acts freely on the ends. In addition, \mathrm{Quinn}(G) acts transitively on those ends which are in one-to-one correspondence with the vertices of G \backslash \mathcal{T} whose stabilizers are isomorphic to \mathrm{GL}_{2}(\mathbb{F}_{q}) .

On w∞ -Warfield Cotorsion Modules and Krull Domains

Let [Formula: see text] be a commutative domain with 1 and [Formula: see text] [Formula: see text] its field of quotients. In this note an [Formula: see text]-module [Formula: see text] is called [Formula: see text]-Warfield cotorsion if [Formula: see text], where [Formula: see text] denotes the class of all Warfield cotorsion [Formula: see text]-modules and [Formula: see text] the class of all [Formula: see text]-projective [Formula: see text]-modules. It is shown that [Formula: see text] is a PVMD if and only if all [Formula: see text]-cotorsion [Formula: see text]-modules are [Formula: see text]-Warfield cotorsion, and that [Formula: see text] is a Krull domain if and only if every [Formula: see text]-Matlis cotorsion strong [Formula: see text]-module over [Formula: see text] is a [Formula: see text]-Warfield cotorsion [Formula: see text]-module.

On quasi-morphic modules

An R-module M is called quasi-morphic if for any f∈EndR(M), there exist g,h∈EndR(M) such that Imf=Kerg and Kerf=Imh. In addition, MR is said to be morphic whenever g=h in the above definition. The main objective of this paper is investigating quasi-morphic property for several classes of modules. First we obtain general properties of quasi-morphic modules via exact sequence approach. Moreover, we investigate conditions under which a finite length quasi-morphic module is morphic. As a result, we show that for uniserial finite length modules, the notions of morphic and quasi-morphic coincide. Over a principal ideal domain R, direct sums of cyclic modules which are (quasi-)morphic are characterized. Among applications of our results, nonsingular extending (quasi-)morphic modules are characterized completely. We also prove that over a commutative Noetherian domain R which is not a field, quasi-morphic nonsingular extending modules are precisely direct sums of copies of Q (the quotient field of R). (Quasi-)Morphic singular extending abelian groups are also characterized.

Integer-valued polynomials on valuation rings of global fields with prescribed lengths of factorizations

Let V be a valuation ring of a global field K. We show that for all positive integers k and 1 < n_1 le cdots le n_k there exists an integer-valued polynomial on V, that is, an element of {{,textrm{Int},}}(V) = { f in K[X] mid f(V) subseteq V }, which has precisely k essentially different factorizations into irreducible elements of {{,textrm{Int},}}(V) whose lengths are exactly n_1,ldots ,n_k. In fact, we show more, namely that the same result holds true for every discrete valuation domain V with finite residue field such that the quotient field of V admits a valuation ring independent of V whose maximal ideal is principal or whose residue field is finite. If the quotient field of V is a purely transcendental extension of an arbitrary field, this property is satisfied. This solves an open problem proposed by Cahen, Fontana, Frisch and Glaz in these cases.

The construction D + DX + DX2 + ⋯ + DXk + Xk+1K[X

Let [Formula: see text] be a [Formula: see text]-Dedekind domain with quotient field [Formula: see text] and [Formula: see text] be a fixed positive integer, it is proved in this paper that the constructed polynomial domain [Formula: see text] is a [Formula: see text]-Prüfer domain. It is also proved that if [Formula: see text] is a principal ideal domain and [Formula: see text], then every maximal ideal of [Formula: see text] is [Formula: see text]-projective.

The construction D + XDS [X] and G-Prüfer domains

Let D be an integral domain with quotient field K, S a (not necessarily saturated) multiplicative subset of D and . In this paper, we show that is a G-Prüfer domain if and only if D is a G-Prüfer domain and DS = K. We also prove that if D is a Bass domain (or a local NWF domain), then every finitely generated torsion-free module over is isomorphic to a direct sum of some ideals.

The center and invariants of standard filiform Lie algebras

This paper describes the centers of the universal enveloping algebras and the invariant rings of the standard filiform Lie algebras over fields of characteristic zero and also over large enough prime characteristic. We determine explicit generators for the quotient fields and also a compact form for the generators for the invariants rings. We prove several combinatorial results concerning the Hilbert series of these algebras.

Polynomial extensions of Krull modules

Let M be a torsion-free module over an integral domain D with quotient field K. In this paper, we show that M is a Krull module over D in the sense of Costa-Johnson (Inert extensions of Krull domains, 1976) if and only if the polynomial module is a Krull module over in the sense of Costa-Johnson.

Polynomials taking integer values on primes in a function field

Let $\mathbb{F}_q[x]$ be the ring of polynomials over a finite field $\mathbb{F}_q$ and $\mathbb{F}_q(x)$ its quotient field. Let $\mathbb{P}$ be the set of primes in $\mathbb{F}_q[x]$, and let $\mathcal{I}$ be the set of all polynomials $f$ over $\mathbb{F}_q(x)$ for which $f(\mathbb{P})\subseteq\mathbb{F}_q[x]$. The existence of a basis for $\mathcal{I}$ is established using the notion of characteristic ideal; this shows that $\mathcal{I}$ is a free $\mathbb{F}_q[x]$-module. Through localization, explicit shapes of certain bases for the localization of $\mathcal{I}$ are derived, and a well-known procedure is described as to how to obtain explicit forms of some bases of $\mathcal{I}$.

On maximal ideals of the polynomial ring and some conjectures on Ext-index of rings

Recall that a ring [Formula: see text] is a Hilbert ring if any maximal ideal of [Formula: see text] contracts to a maximal ideal of [Formula: see text]. The main purpose of this paper is to characterize the prime ideals of a commutative ring [Formula: see text] which are traces of the maximal ideals of the polynomial ring [Formula: see text]. In this context, we prove that if [Formula: see text] is a prime ideal of [Formula: see text] such that [Formula: see text] is a semi-local domain of (Krull) dimension [Formula: see text], then [Formula: see text] is the trace of a maximal ideal of [Formula: see text]. Whereas, if [Formula: see text] is Noetherian and either ([Formula: see text]) or (the quotient field of [Formula: see text] is algebraically closed, [Formula: see text] and [Formula: see text] is not semi-local), then [Formula: see text] is never the trace of a maximal ideal of [Formula: see text]. Putting these results into use in investigating the Ext-index of Noetherian rings, we establish connections between the finiteness of the Ext-index of localizations of the polynomial rings [Formula: see text] and the finiteness of the Ext-index of localizations of [Formula: see text]. This allows us to provide a new class of rings satisfying some known conjectures on Ext-index of Noetherian rings as well as to build bridges between these conjectures.

Prüfer rings in a certain pullback

Let D be an integral domain with quotient field K, X be an indeterminate over D, be the polynomial ring over K, be an integer, , where , and , i.e, . Then Rn is a subring of with total quotient ring . In this paper, we study several ring-theoretic properties of Rn , with a focus on Prüfer rings and Prüfer v-multiplication rings (PvMRs). For example, we show when Rn is a Prüfer ring, a Bézout ring, a PvMR, or a GCD ring.

On periodic stable Auslander–Reiten components containing Heller lattices over the symmetric Kronecker algebra

Let O be a complete discrete valuation ring, K its quotient field, and A the symmetric Kronecker algebra over O. We consider the full subcategory of the category of A-lattices whose objects are A-lattices M such that M⊗OK is projective A⊗OK-modules. In this paper, we study Heller lattices of indecomposable periodic modules over A. As a main result, we determine the shapes of stable Auslander–Reiten components containing Heller lattices of indecomposable periodic modules over A.

Using Semicontinuity for Standard Bases Computations

We present new results on standard basis computations of a 0-dimensional ideal I in a power series ring or in the localization of a polynomial ring over a computable field K. We prove the semicontinuity of the “highest corner” in a family of ideals, parametrized by the spectrum of a Noetherian domain A. This semicontinuity is used to design a new modular algorithm for computing a standard basis of I if K is the quotient field of A. It uses the computation over the residue field of a “good” prime ideal of A to truncate high order terms in the subsequent computation over K. We prove that almost all prime ideals are good, so a random choice is very likely to be good, and whether it is good is detected a posteriori by the algorithm. The algorithm yields a significant speed advantage over the non-modular version and works for arbitrary Noetherian domains. The most important special cases are perhaps A={mathbb {Z}} and A=k[t], k any field and t a set of parameters. Besides its generality, the method differs substantially from previously known modular algorithms for A={mathbb {Z}}, since it does not manipulate the coefficients. It is also usually faster and can be combined with other modular methods for computations in local rings. The algorithm is implemented in the computer algebra system Singular and we present several examples illustrating its power.

A note on integer-valued skew polynomials

Given an integral domain [Formula: see text] with quotient field [Formula: see text], the study of the ring of integer-valued polynomials [Formula: see text] has attracted a lot of attention over the past decades. Recently, Werner has extended this study to the situation of skew polynomials. To be more precise, if [Formula: see text] is an automorphism of [Formula: see text], one may consider the set [Formula: see text], where [Formula: see text] is the skew polynomial ring and [Formula: see text] is a “suitable” evaluation of [Formula: see text] at [Formula: see text]. For example, he gave sufficient conditions for [Formula: see text] to be a ring and study some of its properties. In this paper, we extend the study to the situation of the skew polynomial ring [Formula: see text] with a suitable evaluation, where [Formula: see text] is a [Formula: see text]-derivation. Moreover we prove, for example, that if [Formula: see text] is of finite order and [Formula: see text] is a Dedekind domain with finite residue fields such that [Formula: see text] is a ring, then [Formula: see text] is non-Noetherian.

Operational Calculus for the General Fractional Derivatives of Arbitrary Order

In this paper, we deal with the general fractional integrals and the general fractional derivatives of arbitrary order with the kernels from a class of functions that have an integrable singularity of power function type at the origin. In particular, we introduce the sequential fractional derivatives of this type and derive an explicit formula for their projector operator. The main contribution of this paper is a construction of an operational calculus of Mikusiński type for the general fractional derivatives of arbitrary order. In particular, we present a representation of the m-fold sequential general fractional derivatives of arbitrary order as algebraic operations in the field of convolution quotients and derive some important operational relations.