On automorphisms of unique factorization domains

1985 ◽  
Vol 98 (3) ◽  
pp. 427-428
Author(s):  
M. L. Brown
Keyword(s):  
Finite Field ◽  
Finite Order ◽  
Residue Field ◽  
Prime Ideals ◽  
Unique Factorization ◽  
Unique Factorization Domain ◽  
Unique Factorization Domains

Chatters has asked whether a unique factorization domain (UFD) R, equipped with an automorphism σ transitive on the height 1 prime ideals of R, is necessarily Dedekind. If R contains an uncountable field, then Chatters observed that the answer is affirm ative. In this note was show:Theorem. Let R be a local UFD equipped with an automorphism σ which is transitive on the height 1 prime ideals of R. Suppose that σ induces an automorphism of finite order on the residue field κ of R (for example, if κ is a global or finite field), Then R is Dedekind.

Formal Fibers of Unique Factorization Domains

2010 ◽  
Vol 62 (4) ◽  
pp. 721-736 ◽  
Author(s):  
Adam Boocher ◽  
Michael Daub ◽  
Ryan K. Johnson ◽  
H. Lindo ◽  
S. Loepp ◽  
...  
Keyword(s):  
Noetherian Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Prime Ideals ◽  
Unique Factorization ◽  
Tight Closure ◽  
Unique Factorization Domain ◽  
Necessary And Sufficient ◽  
Unique Factorization Domains ◽  
Adic Completion

AbstractLet (T,M) be a complete local (Noetherian) ring such that dimT ≥ 2 and |T| = |T/M| and let ﹛pi﹜i∈𝒥 be a collection of elements of T indexed by a set I so that |𝒥| < |T|. For each i ∈ 𝒥, let Ci := ﹛Qi1, … ,Qini ﹜ be a set of nonmaximal prime ideals containing pi such that the Qi j are incomparable and pi ∈ Qjk if and only if i = j. We provide necessary and sufficient conditions so that T is the m -adic completion of a local unique factorization domain (A,m ), and for each i ∈ I, there exists a unit ti of T so that pi ti ∈ A andCi is the set of prime ideals Q of T that are maximal with respect to the condition that Q ∩ A = pi tiA.We then use this result to construct a (nonexcellent) unique factorization domain containingmany ideals for which tight closure and completion do not commute. As another application, we construct a unique factorization domain A most of whose formal fibers are geometrically regular.

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

scholarly journals On Zero Left Prime Factorizations for Matrices over Unique Factorization Domains

2020 ◽  
Vol 2020 ◽  
pp. 1-3
Author(s):  
Jinwang Liu ◽  
Tao Wu ◽  
Dongmei Li ◽  
Jiancheng Guan
Keyword(s):  
Unique Factorization ◽  
Unique Factorization Domain ◽  
Class Of Matrices ◽  
Unique Factorization Domains

In this paper, zero prime factorizations for matrices over a unique factorization domain are studied. We prove that zero prime factorizations for a class of matrices exist. Also, we give an algorithm to directly compute zero left prime factorizations for this class of matrices.

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.

Eisenstein's Criteria for Absolute Irreducibility Over a Finite Field

1966 ◽  
Vol 9 (05) ◽  
pp. 575-580 ◽  
Author(s):  
Kenneth S. Williams
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Galois Field ◽  
Unique Factorization ◽  
Unique Factorization Domain ◽  
Irreducibility Criteria

Let p denote a prime and n a positive integer. Write q = pn and let kq denote the Galois field with q elements. The unique factorization domain of polynomials in m(≤ 2) indeterminâtes x1,…, xq with coefficients in k is denoted by kq [x,…, xm. It is the purpose of this note to prove the foliowing generalization of Eisenstein's irreducibility criteria and to point out some of its consequences.

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 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.

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