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.

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.

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.

Additive decompositions of polynomials over unique factorization domains

2019 ◽  
Vol 19 (08) ◽  
pp. 2050150
Author(s):  
Leila Benferhat ◽  
Safia Manar Elislam Benoumhani ◽  
Rachid Boumahdi ◽  
Jesse Larone
Keyword(s):  
Finite Fields ◽  
Unique Factorization ◽  
Additive Decomposition ◽  
Unique Factorization Domains

Additive decompositions over finite fields were extensively studied by Brawely and Carlitz. In this paper, we study the additive decomposition of polynomials over unique factorization domains.

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.

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.

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