ON BOUNDING MEASURES OF PRIMENESS IN INTEGRAL DOMAINS

2012 ◽  
Vol 22 (05) ◽  
pp. 1250040 ◽  
Author(s):  
DAVID F. ANDERSON ◽  
SCOTT T. CHAPMAN
Keyword(s):  
Integral Domain ◽  
Unique Factorization ◽  
Integral Domains ◽  
Unique Factorization Domain ◽  
The Relationship

Let D be an integral domain. In this paper, we investigate two (integer- or ∞-valued) invariants ω(D, x) and ω(D) which measure how far a nonzero x ∈ D is from being prime and how far an atomic integral domain D is from being a unique factorization domain (UFD), respectively. In particular, we are interested in when there is a nonzero (irreducible) x ∈ D with ω(D, x) = ∞ and the relationship between ω(A, x) and ω(B, x), and ω(A) and ω(B), for an extension A ⊆ B of integral domains and a nonzero x ∈ A.

On a Class of Archimedean Integral Domains

1976 ◽  
Vol 28 (2) ◽  
pp. 365-375 ◽  
Author(s):  
Raymond A. Beauregard ◽  
David E. Dobbs
Keyword(s):  
Number Theory ◽  
Integral Domain ◽  
Unique Factorization ◽  
Elementary Number Theory ◽  
Integral Domains ◽  
Elementary Number ◽  
Unique Factorization Domain ◽  
Starting Point ◽  
Positive Integers

Our starting point is an observation in elementary number theory [10, Exercise 26, p. 17]: if a and b are positive integers such that each number in the sequence a, b2, a3, b4, … divides the next, then a = b. Its proof depends only on Z being a unique factorization domain (UFD) whose units are 1, —1. Accordingly, we abstract and say that a (commutative integral) domain R satisfies (*) in case, whenever nonzero elements a and b in R are such that each element in the sequence a, b2, a3, b4, … divides the next, then a and b are associates in R (that is, a = bu for some unit u of R). The main objective of this paper is the study of the class of domains satisfying (*).

An Analog of Nagata's Theorem for Modular LCM Domains

1977 ◽  
Vol 29 (2) ◽  
pp. 307-314 ◽  
Author(s):  
Raymond A. Beauregard
Keyword(s):  
Integral Domain ◽  
Quotient Ring ◽  
Unique Factorization ◽  
Unique Factorization Domain

The theorem referred to in the title asserts that for an atomic commutative integral domain R, if S is a submonoid of R* (the monoid of nonzero elements of R) generated by primes such that the quotient ring RS-1 is a UFD (unique factorization domain) then R is also a UFD [8]. Recently several definitions of a noncommutative UFD have been proposed (see the summary in [6]).

scholarly journals π-domains, overrings, and divisorial ideals

1978 ◽  
Vol 19 (2) ◽  
pp. 199-203 ◽  
Author(s):  
D. D. Anderson
Keyword(s):  
Principal Ideal ◽  
Integral Domain ◽  
Prime Ideals ◽  
Unique Factorization ◽  
Unique Factorization Domain ◽  
Prufer Domains ◽  
Interesting Class ◽  
Prüfer Domains ◽  
Krull Domains

In this paper we study several generalizations of the concept of unique factorization domain. An integral domain is called a π-domain if every principal ideal is a product of prime ideals. Theorem 1 shows that the class of π-domains forms a rather natural subclass of the class of Krull domains. In Section 3 we consider overrings of π-domains. In Section 4 generalized GCD-domains are introduced: these form an interesting class of domains containing all Prüfer domains and all π-domains.

Unique Factorization Theorems for Subalgebras of the Incidence Algebra

1972 ◽  
Vol 24 (5) ◽  
pp. 967-977
Author(s):  
K. L. Yocom
Keyword(s):  
Partially Ordered Set ◽  
Integral Domain ◽  
Ordered Set ◽  
Sufficient Conditions ◽  
Unique Factorization ◽  
Countable Number ◽  
Incidence Algebra ◽  
Direct Sum ◽  
Unique Factorization Domain ◽  
Necessary And Sufficient

H. Scheid [4] has found necessary and sufficient conditions on a partially ordered set S(≦) which is a direct sum of a countable number of trees for a certain subalgebra G(+, *) of the incidence algebra F(+, *) to be an integral domain. In this paper we prove that under similar conditions on S, G(+, *) is actually a unique factorization domain or, failing this, that there is a subalgebra H(+, *) of F(+, *) which is a unique factorization domain and contains G. Similar results are then obtained as corollaries in the regular convolution rings of Narkiewicz.

Unique factorization rings

1969 ◽  
Vol 65 (3) ◽  
pp. 579-583 ◽  
Author(s):  
C. R. Fletcher
Keyword(s):  
Commutative Ring ◽  
Unique Factorization ◽  
Integral Domains ◽  
Zero Divisors ◽  
Unique Factorization Domain

1. The concept of a unique factorization domain (UFD) has been defined, for commutative (e.g. (4) page 21) and non-commutative (1) integral domains. We take the theory a stage further here by defining a unique factorization ring (UFR), where throughout, a ring is understood to mean a commutative ring with identity, possibly containing proper zero-divisors.

scholarly journals Elasticity in Polynomial-Type Extensions

2016 ◽  
Vol 59 (3) ◽  
pp. 581-590 ◽  
Author(s):  
Mark Batell ◽  
Jim Coykendall
Keyword(s):  
Polynomial Ring ◽  
Integral Domain ◽  
Irreducible Polynomial ◽  
Polynomial Rings ◽  
Sufficient Condition ◽  
Integral Domains ◽  
Polynomial Type ◽  
Factorial Domain ◽  
The Relationship

AbstractThe elasticity of an atomic integral domain is, in some sense, a measure of how far the domain is from being a half-factorial domain. We consider the relationship between the elasticity of a domain R and the elasticity of its polynomial ring R[x]. For example, if R has at least one atom, a sufficient condition for the polynomial ring R[x] to have elasticity 1 is that every non-constant irreducible polynomial f ∈ R[x] be irreducible in K[x]. We will determine the integral domains R whose polynomial rings satisfy this condition.

scholarly journals Decomposition and classification of length functions

2020 ◽  
Vol 32 (5) ◽  
pp. 1109-1129
Author(s):  
Dario Spirito
Keyword(s):  
Integral Domain ◽  
Integral Domains ◽  
Bijective Correspondence ◽  
Length Functions ◽  
Standard Representation ◽  
Prufer Domains ◽  
Prüfer Domains

AbstractWe study decompositions of length functions on integral domains as sums of length functions constructed from overrings. We find a standard representation when the integral domain admits a Jaffard family, when it is Noetherian and when it is a Prüfer domains such that every ideal has only finitely many minimal primes. We also show that there is a natural bijective correspondence between singular length functions and localizing systems.

Graded integral domains in which each nonzero homogeneous t-ideal is divisorial

2019 ◽  
Vol 18 (01) ◽  
pp. 1950018 ◽  
Author(s):  
Gyu Whan Chang ◽  
Haleh Hamdi ◽  
Parviz Sahandi
Keyword(s):  
Integral Domain ◽  
Nonzero Ideal ◽  
Integral Domains ◽  
Cancellative Monoid ◽  
Integrally Closed ◽  
Graded Integral Domain

Let [Formula: see text] be a nonzero commutative cancellative monoid (written additively), [Formula: see text] be a [Formula: see text]-graded integral domain with [Formula: see text] for all [Formula: see text], and [Formula: see text]. In this paper, we study graded integral domains in which each nonzero homogeneous [Formula: see text]-ideal (respectively, homogeneous [Formula: see text]-ideal) is divisorial. Among other things, we show that if [Formula: see text] is integrally closed, then [Formula: see text] is a P[Formula: see text]MD in which each nonzero homogeneous [Formula: see text]-ideal is divisorial if and only if each nonzero ideal of [Formula: see text] is divisorial, if and only if each nonzero homogeneous [Formula: see text]-ideal of [Formula: see text] is divisorial.

scholarly journals On the Graph of Divisibility of an Integral Domain

2015 ◽  
Vol 58 (3) ◽  
pp. 449-458 ◽  
Author(s):  
Jason Greene Boynton ◽  
Jim Coykendall
Keyword(s):  
Topological Space ◽  
Integral Domain ◽  
Point Of View ◽  
Current Understanding ◽  
Main Theme ◽  
Topological Graph ◽  
Integral Domains ◽  
Graph Theoretic ◽  
Theoretic Point ◽  
Group Of Divisibility

AbstractIt is well known that the factorization properties of a domain are reflected in the structure of its group of divisibility. The main theme of this paper is to introduce a topological/graph-theoretic point of view to the current understanding of factorization in integral domains. We also show that connectedness properties in the graph and topological space give rise to a generalization of atomicity.

Algebraic Homotopy Theory

1981 ◽  
Vol 33 (2) ◽  
pp. 302-319 ◽  
Author(s):  
J. F. Jardine
Keyword(s):  
Homotopy Type ◽  
Homotopy Theory ◽  
Unique Factorization ◽  
Contravariant Functor ◽  
Unique Factorization Domain ◽  
Simplicial Sets ◽  
Simplicial Set ◽  
Image Position ◽  
Algebraic Homotopy Theory

Kan and Miller have shown in [9] that the homotopy type of a finite simplicial set K can be recovered from its R-algebra of 0-forms A0K, when R is a unique factorization domain. More precisely, if is the category of simplicial sets and is the category of R-algebras there is a contravariant functorwiththe simplicial set homomorphisms from X to the simplicial R-algebra ∇, whereand the faces and degeneracies of ∇ are induced byandrespectively.

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