An application of local uniformization to the theory of divisors

1951 ◽  
Vol 47 (2) ◽  
pp. 279-285
Author(s):  
D. G. Northcott
Keyword(s):  
Quotient Ring ◽  
Local Rings ◽  
Simple Point ◽  
Unique Factorization ◽  
The Third ◽  
Unique Factorization Domain ◽  
Irreducible Variety ◽  
Structure Theorems ◽  
Local Uniformization

If V is an irreducible variety and W is an irreducible simple subvariety of V, then one of the properties of the quotient ring of W in V is that it is a unique factorization domain. A proof of this theorem has been given by Zariski ((2), Theorem 5, p. 22), based on the structure theorems for complete local rings, and the fact that the local rings which arise geometrically are always analytically unramified. Here the theorem is deduced from certain properties of functions and their divisors which will be established by entirely different considerations. The terminology which will be employed is that proposed by A. Weil in his book(1), and we shall use, for instance, F-viii, Th. 3, Cor. 1, when referring to Corollary 1 of the third theorem in Chapter 8. Before proceeding to details it should be noted that Weil and Zariski differ in then-definitions, and that in particular the terms ‘variety’ and ‘simple point’ do not mean quite the same in the two theories. The effect of this is to make Zariski's result somewhat stronger than Theorem 3 of this paper.

scholarly journals Hereditary Local Rings

1966 ◽  
Vol 27 (1) ◽  
pp. 223-230 ◽  
Author(s):  
P. M. Cohn
Keyword(s):  
Local Ring ◽  
Local Rings ◽  
Unique Factorization ◽  
Prime Factorization ◽  
Unique Factorization Domain ◽  
Necessary And Sufficient

Many questions about free ideal rings ( = firs, cf. [5] and §2 below) which at present seem difficult become much easier when one restricts attention to local rings. One is then dealing with hereditary local rings, and any such ring is in fact a fir (§2). Our object thus is to describe hereditary local rings. The results on firs in [5] show that such a ring must be a unique factorization domain; in §3 we prove that it must also be rigid (cf. the definition in [3] and §3 below). More precisely, for a semifir R with prime factorization rigidity is necessary and sufficient for R to be a local ring.

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]).

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.

scholarly journals Unique factorization in polynomial rings with zero divisors

2020 ◽  
pp. 2150113 ◽  
Author(s):  
D. D. Anderson ◽  
Ranthony A. C. Edmonds
Keyword(s):  
Commutative Ring ◽  
Polynomial Ring ◽  
Commutative Rings ◽  
Polynomial Rings ◽  
Unique Factorization ◽  
Factorization Property ◽  
Zero Divisors ◽  
Unique Factorization Domain

Given a certain factorization property of a ring [Formula: see text], we can ask if this property extends to the polynomial ring over [Formula: see text] or vice versa. For example, it is well known that [Formula: see text] is a unique factorization domain if and only if [Formula: see text] is a unique factorization domain. If [Formula: see text] is not a domain, this is no longer true. In this paper, we survey unique factorization in commutative rings with zero divisors, and characterize when a polynomial ring over an arbitrary commutative ring has unique factorization.

scholarly journals Monoidal extensions of a Cohen-Macaulay unique factorization domain

2002 ◽  
Vol 354 (5) ◽  
pp. 1811-1835 ◽  
Author(s):  
William J. Heinzer ◽  
Aihua Li ◽  
Louis J. Ratliff ◽  
David E. Rush
Keyword(s):  
Unique Factorization ◽  
Unique Factorization Domain

scholarly journals Unique factorization in the ring R[x]

1992 ◽  
Vol 53 (3) ◽  
pp. 287-293 ◽  
Author(s):  
Raymond A. Beauregard
Keyword(s):  
Left Ideal ◽  
Polynomial Ring ◽  
The Other ◽  
Unique Factorization ◽  
Ideal Domain ◽  
Unique Factorization Domain ◽  
Other Hand ◽  
Reasonable Sense

AbstractIf R is a commutative unique factorization domain (UFD) then so is the ring R[x]. If R is not commutative then no such result is possible. An example is given of a bounded principal right and left ideal domain R, hence a similarity-UFD, for which the polynomial ring R[x] in a central indeterminate x is not a UFD in any reasonable sense. On the other hand, it is shown that if R is an invariant UFD then R[x] is a UFD in an appropriate sense.

scholarly journals Controlling the dimensions of formal fibers of a unique factorization domain at the height one prime ideals

2018 ◽  
Vol 10 (4) ◽  
pp. 475-498
Author(s):  
Sarah M. Fleming ◽  
Lena Ji, S. Loepp ◽  
Peter M. McDonald ◽  
Nina Pande ◽  
David Schwein
Keyword(s):  
Prime Ideals ◽  
Unique Factorization ◽  
Unique Factorization Domain

The Complete Quotient Ring of Images of Semilocal Prüfer Domains

1977 ◽  
Vol 29 (5) ◽  
pp. 914-927 ◽  
Author(s):  
John Chuchel ◽  
Norman Eggert
Keyword(s):  
Homomorphic Image ◽  
Noetherian Ring ◽  
Quotient Ring ◽  
Local Rings ◽  
Topological Ring ◽  
Prüfer Domain ◽  
Prufer Domains ◽  
Prüfer Domains ◽  
Quotient Rings ◽  
Prüfer Rings

It is well known that the complete quotient ring of a Noetherian ring coincides with its classical quotient ring, as shown in Akiba [1]. But in general, the structure of the complete quotient ring of a given ring is largely unknown. This paper investigates the structure of the complete quotient ring of certain Prüfer rings. Boisen and Larsen [2] considered conditions under which a Prüfer ring is a homomorphic image of a Prüfer domain and the properties inherited from the domain. We restrict our investigation primarily to homomorphic images of semilocal Prüfer domains. We characterize the complete quotient ring of a semilocal Prüfer domain in terms of complete quotient rings of local rings and a completion of a topological ring.

Hilbert rings and G(oldman)-rings issued from amalgamated algebras

2018 ◽  
Vol 17 (02) ◽  
pp. 1850023 ◽  
Author(s):  
L. Izelgue ◽  
O. Ouzzaouit
Keyword(s):  
New York ◽  
Commutative Algebra ◽  
Quotient Ring ◽  
Sufficient Conditions ◽  
Ring Homomorphism ◽  
Necessary And Sufficient Conditions ◽  
Local Rings ◽  
Finitely Generated ◽  
Necessary And Sufficient ◽  
The Way

Let [Formula: see text] and [Formula: see text] be two rings, [Formula: see text] an ideal of [Formula: see text] and [Formula: see text] be a ring homomorphism. The ring [Formula: see text] is called the amalgamation of [Formula: see text] with [Formula: see text] along [Formula: see text] with respect to [Formula: see text]. It was proposed by D’anna and Fontana [Amalgamated algebras along an ideal, Commutative Algebra and Applications (W. de Gruyter Publisher, Berlin, 2009), pp. 155–172], as an extension for the Nagata’s idealization, which was originally introduced in [Nagata, Local Rings (Interscience, New York, 1962)]. In this paper, we establish necessary and sufficient conditions under which [Formula: see text], and some related constructions, is either a Hilbert ring, a [Formula: see text]-domain or a [Formula: see text]-ring in the sense of Adams [Rings with a finitely generated total quotient ring, Canad. Math. Bull. 17(1) (1974)]. By the way, we investigate the transfer of the [Formula: see text]-property among pairs of domains sharing an ideal. Our results provide original illustrating examples.

scholarly journals Arithmetic progressions in a unique factorization domain

2012 ◽  
Vol 154 (2) ◽  
pp. 161-171
Author(s):  
Sudhir R. Ghorpade ◽  
Samrith Ram
Keyword(s):  
Arithmetic Progressions ◽  
Unique Factorization ◽  
Unique Factorization Domain
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