Factorial Domain Research Articles

Locally nilpotent derivations of factorial domains

Let [Formula: see text] be factorial domains containing [Formula: see text]. In this paper, we give a criterion, in terms of locally nilpotent derivations, for [Formula: see text] to be [Formula: see text]-isomorphic to [Formula: see text], where [Formula: see text] is nonzero and [Formula: see text]. As a consequence, we retrieve a recent result due to Masuda [Families of hypersurfaces with noncancellation property, Proc. Amer. Math. Soc. 145(4) (2017) 1439–1452] characterizing Danielewski hypersurfaces whose coordinate ring is factorial. We also apply our criterion to the study of triangularizable locally nilpotent [Formula: see text]-derivations of the polynomial ring in two variables over [Formula: see text].

Semigroup rings as weakly factorial domains, II

Let [Formula: see text] be an integral domain, [Formula: see text] be a nonzero torsionless commutative cancellative monoid with quotient group [Formula: see text], and [Formula: see text] be the semigroup ring of [Formula: see text] over [Formula: see text]. In this paper, among other things, we show that if [Formula: see text] (respectively, [Formula: see text], then [Formula: see text] is a weakly factorial domain if and only if [Formula: see text] is a weakly factorial GCD-domain, [Formula: see text] is a weakly factorial GCD-semigroup, and [Formula: see text] is of type [Formula: see text] (respectively, [Formula: see text] except [Formula: see text]).

Weakly factorial property of a generalized Rees ring $D[X,d/X

Let $D$ be an integral domain, $X$ an indeterminate over $D$, $d \in D$, and $R = D[X, {d}/{X}]$ a subring of $D[X,{1}/{X}]$. In this paper, we show that $R$ is a weakly factorial domain if and only if $D$ is a weakly factorial GCD-domain and $d=0$, $d$ is a unit of $D$ or $d$ is a prime element of $D$. We also show that, if $D$ is a weakly factorial GCD-domain, $p$ is a prime element of $D$, and $n \geq 2$ is an integer, then $D[X, {p^n}/{X}]$ is an almost weakly factorial domain with $Cl(D[X, {p^n}/{X}]) = \mathbb {Z}_n$.

An extension of a theorem of Frobenius and Stickelberger to modules of projective dimension one over a factorial domain

Let R be a Cohen–Macaulay ring. A quasi-Gorenstein R-module is an R-module such that the grade of the module and the projective dimension of the module are equal and the canonical module of the module is isomorphic to the module itself. After discussing properties of finitely generated quasi-Gorenstein modules, it is shown that this definition allows for a characterization of diagonal matrices of maximal rank over a Cohen–Macaulay factorial domain R extending a theorem of Frobenius and Stickelberger to modules of projective dimension 1 over a commutative factorial Cohen–Macaulay domain.

Recent results on weakly factorial domains

In this paper, we will survey recent results on weakly factorial domains base on the results of [11, 13, 14]. LetD be an integral domain, X be an indeterminate over D, d ∈ D, R = D[X,d/X] be a subring of the Laurent polynomial ring D[X,1/X], Γ be a nonzero torsionless commutative cancellative monoid with quotient group G, and D[Γ] be the semigroup ring of Γ over D. Among other things, we show that R is a weakly factorial domain if and only if D is a weakly factorial GCD‐domain and d = 0, d is a unit of D or d is a prime element of D. We also show that if char(D) = 0 (resp., char(D) = p > 0), then D[Γ] is a weakly factorial domain if and only if D is a weakly factorial GCD domain, Γ is a weakly factorial GCD semigroup, and G is of type (0,0,0,…) (resp., (0,0,0,…) except p).

ON THE DIVISOR-CLASS GROUP OF MONADIC SUBMONOIDS OF RINGS OF INTEGER-VALUED POLYNOMIALS

Let $R$ be a factorial domain. In this work we investigate the connections between the arithmetic of ${\rm Int}(R)$ (i.e., the ring of integer-valued polynomials over $R$) and its monadic submonoids (i.e., monoids of the form $\{g\in {\rm Int}(R)\mid g\mid_{{\rm Int}(R)} f^k$ for some $k\in\mathbb{N}_0\}$ for some nonzero $f\in {\rm Int}(R)$). Since every monadic submonoid of ${\rm Int}(R)$ is a Krull monoid it is possible to describe the arithmetic of these monoids in terms of their divisor-class group. We give an explicit description of these divisor-class groups in several situations and provide a few techniques that can be used to determine them. As an application we show that there are strong connections between ${\rm Int}(R)$ and its monadic submonoids. If $R=\mathbb{Z}$ or more generally if $R$ has sufficiently many nice atoms, then we prove that the infinitude of the elasticity and the tame degree of ${\rm Int}(R)$ can be explained by using the structure of monadic submonoids of ${\rm Int}(R)$.

Radical perfectness of prime ideals in certain integral domains

For a UMT-domain $D$, we characterize when the polynomial ring $D[X]$ is $t$-compactly packed and every prime $t$-ideal of $D[X]$ is radically perfect. As a corollary, for a quasi-Pr\"ufer domain $D$, we also characterize when every prime ideal of $D[X]$ is radically perfect. Finally we introduce the concepts of Serre's conditions in strong Mori domains and characterize Krull domains and almost factorial domains, respectively.

Half-factorial subrings of factorial domains

A half-factorial domain D is a domain in which every non-zero element that is not a unit is a product of a unique number of irreducible elements of D. We characterize half-factorial subrings R of factorial domains S when S is the integral closure of R and their unit groups are identical. Let A be a factorial domain and A[T] the polynomial ring over A in the variable T. The characterization is used to describe the half-factorial A-subalgebras R with multiplicative conductors of A[T] into R.

Triangular systems and a generalization of primitive polynomials

A well-known generalization of the notion of primitive polynomials over factorial domains to arbitrary commutative rings is analyzed and applied to triangular systems. In particular, we clarify when a triangular system generates its saturation ideal, thus correcting a criterion published first in 2008.

A General Theory of Almost Splitting Sets

Let * be a star-operation of finite type on an integral domain D. In this paper, we generalize and study the concept of almost splitting sets. We define a saturated multiplicative subset S of D to be an almost g*-splitting set of D if for each 0 ≠ d ∈ D, there exists an integer n = n(d) ≥1 such that d n = st for some s ∈ S and t ∈ D with (t, s′)* = D for all s′ ∈ S. Among other things, we prove that every saturated multiplicative subset of D is an almost g*-splitting set if and only if D is an almost weakly factorial domain (AWFD) with *-dim (D) = 1. We also give an example of an almost g*-splitting set which is not a g*-splitting set.

T-SPLITTING SETS S OF AN INTEGRAL DOMAIN D SUCH THAT DSIS A FACTORIAL DOMAIN

Let $D$ be an integral domain, $S$ be a saturated multiplicative subset of $D$ such that $D_S$ is a factorial domain, $\{X_{\alpha}\}$ be a nonempty set of indeterminates, and $D[\{X_{\alpha}\}]$ be the polynomial ring over $D$. We show that $S$ is a splitting (resp., almost splitting, $t$-splitting) set in $D$ if and only if every nonzero prime $t$-ideal of $D$ disjoint from $S$ is principal (resp., contains a primary element, is $t$-invertible). We use this result to show that $D \setminus \{0\}$ is a splitting (resp., almost splitting, $t$-splitting) set in $D[\{X_{\alpha}\}]$ if and only if $D$ is a GCD-domain (resp., UMT-domain with $Cl(D[\{X_{\alpha}\}])$ torsion, UMT-domain).

Weakly Krull domains and the composite numerical semigroup ringD+E[Γ∗

Let D E be an extension of integral domains, 0 a numerical semigroup with 0 ( N0, 0 D 0nf0g and RD DC ET0 U. In this paper, we completely characterize when R is a weakly Krull domain, an AWFD or a GWFD. We also prove that R is never a WFD. Let T.D/ be the abelian group of t-invertible fractional t-ideals of D under the t- multiplication I JD.I J/t , and let Inv.D/ and Prin.D/ be the subgroups of T.D/ consisting respectively of invertible fractional ideals of D and nonzero principal fractional ideals of D. Then it is clear that Prin.D/ Inv.D/ T.D/. The t-class group of D is an abelian group Cl.D/D T.D/= Prin.D/ and the Picard group Pic.D/D Inv.D/= Prin.D/ is a subgroup of Cl.D/. The local t-class group G.D/ of D is defined by G.D/D Cl.D/= Pic.D/. Let X 1 .D/ stand for the set of height-one prime ideals of D. We say that D is a weakly Krull domain if DD T P2X 1 .D/ DP and this intersection has finite character, i.e., each nonzero element d2 D is a unit in DP for all but a finite number of P's in X 1 .D/; D is a weakly factorial domain (WFD) if every nonzero nonunit element of D is a product of primary elements; D is an almost weakly factorial domain

Some patching results on algebras over two-dimensional factorial domains

Let R=k[x,y] be the polynomial ring in two variables over a field k. We investigate the structure and properties of R-algebras A which are obtained as A=Ax∩Ay where Ax and Ay are polynomial algebras in one variable over Rx and Ry respectively. Most of our results hold when R is a two-dimensional UFD and x,y is an R-regular sequence generating a maximal ideal of R.

Semigroup Rings as Weakly Factorial Domains

Let D be an integral domain, Γ be a torsion-free grading monoid with quotient group G, and D[Γ] be the semigroup ring of Γ over D. We show that if G is of type (0, 0, 0,…), then D[Γ] is a weakly factorial domain if and only if D is a weakly factorial GCD-domain and Γ is a weakly factorial GCD-semigroup. Let ℝ be the field of real numbers and Γ be the additive semigroup of nonnegative rational numbers. We also show that Γ is a weakly factorial GCD-semigroup, but ℝ[Γ] is not a weakly factorial domain.

Noncommutative curves of genus zero: related to finite dimensional algebras

In these notes the author investigates noncommutative smooth projective curves of genus zero, also called exceptional curves. As a main result he shows that each such curve X admits, up to some weighting, a projective coordinate algebra which is a not necessarily commutative graded factorial domain R in the sense of Chatters and Jordan. Moreover, there is a natural bijection between the points of X and the homogeneous prime ideals of height one in R, and these prime ideals are principal in a strong sense.

Power series extensions of half-factorial domains

An integral domain is said to be a half-factorial domain (HFD) if every non-zero element a that is not a unit may be factored into a finite product of irreducible elements, while any other such factorization of a has the same number of irreducible factors. While it is known that a power series extension of a factorial domain need not be factorial, the corresponding question for HFD has been open. In this paper we show that the answer is also negative. In the process we answer in the negative, for HFD, an open question of Samuel for factorial domains by showing that for certain quadratic domains R , and independent variables, Y and T , R [ [ Y ] ] [ [ T ] ] is not HFD even when R [ [ Y ] ] is HFD. The proof hinges on Samuel’s theorem to the effect that a power series, in finitely many variables, over a regular factorial domain is factorial.

SPECTRAL LOCALIZING SYSTEMS THAT ARE t-SPLITTING MULTIPLICATIVE SETS OF IDEALS

Let D be an integral domain with quotient field K, A a nonempty set of height-one maximal t-ideals of D, F<TEX>$({\Lambda})={I{\subseteq}D|I$</TEX> is an ideal of D such that <TEX>$I{\subseteq}P$</TEX> for all <TEX>$P{\in}A}$</TEX>, and <TEX>$D_F({\Lambda})={x{\in}K|xA{\subseteq}D$</TEX> for some <TEX>$A{\in}F({\Lambda})}$</TEX>. In this paper, we prove that if each <TEX>$P{\in}A$</TEX> is the radical of a finite type v-ideal (resp., a principal ideal), then <TEX>$D_{F({\Lambda})}$</TEX> is a weakly Krull domain (resp., generalized weakly factorial domain) if and only if the intersection <TEX>$D_{F({\Lambda})}={\cap}_{P{\in}A}D_P$</TEX> has finite character, if and only if <TEX>$F({\Lambda})$</TEX> is a t-splitting set of ideals, if and only if <TEX>$F({\Lambda})$</TEX> is v-finite.

Integral Domains Having Nonzero Elements with Infinitely Many Prime Divisors

In a factorial domain, every nonzero element has only finitely many prime divisors. We study integral domains having nonzero elements with infinitely many prime divisors.

Assessment of a redox alkaline/iron-chelate absorption process for the removal of dilute hydrogen sulfide in air emissions

The oxidative absorption of hydrogen sulfide ( H 2 S ) into a solution of ferric chelate of trans-1,2- diaminocyclohexanetetraacetate (CDTA) was studied in a counter-current laboratory column randomly packed with 15 mm plastic Ralu rings. The present investigation takes concern about the Kraft pulping situation where dilute H 2 S concentrations are omnipresent in large-volume gas effluents. A fractional two-level factorial approach was instigated to determine the significance of six operating variables, namely the solution's alkalinity (pH; 8.5–10.5), the liquid mass flow rate ( L ; 1.73 – 5.19 kg m - 2 s - 1 ), the solution's ionic strength ( I C ; 0.01 – 0.1 mol dm - 3 ), the gas mass flow rate ( G ; 0.19 – 0.57 kg m - 2 s - 1 ), the inlet H 2 S concentration ( C H 2 S , 0 ; 70 –430 ppm) and the initial ferric CDTA concentration ( C Fe , 0 ; 100 – 400 μ mol dm - 3 ). Initially, a Plackett–Burman design matrix of seven duplicated experiments revealed that pH is the leading factor controlling the H 2 S conversion rate while the ionic strength and ferric CDTA concentration effects remained negligible within the factorial domain. Surface response analysis based on 11 duplicated factorial experiments plus 10 central composite trials revealed that the H 2 S conversion significantly increases with liquid flow rate but decreases with growing H 2 S load up. Further examination about the influence of ferric CDTA on H 2 S absorption rate was set up over a broader concentration range ( C Fe , 0 ; 0 – 2000 μ mol dm - 3 ) at pH of 9.5 and 10.5. It showed good potential at 2000 μ mol dm - 3 as H 2 S conversion increased by a significant 25% for both pH values in comparison to pure alkaline solutions containing no ferric CDTA.

When Are There Infinitely Many Irreducible Elements in a Principal Ideal Domain?

It has been a well-known fact since Euclid's time that there exist infinitely many rational primes. Two natural questions arise: In which other rings, sufficiently similar to the integers, are there infinitely many irreducible elements? Is there a unifying algebraic concept that characterizes such rings? The purpose of this note is to place the fact concerning the infinity of primes into a more general context, one that also includes the interesting case of the factorial domains of algebraic integers in a number field. We show that, if $A$ is a P.I.D., then $A$ contains infinitely many (pairwise nonassociate) irreducible elements if and only if every maximal ideal of $A[x]$ has the same (maximal) height.