scholarly journals Prime ideals in skew Laurent polynomial rings

1993 ◽  
Vol 36 (2) ◽  
pp. 299-317 ◽  
Author(s):  
K. W. Mackenzie
Keyword(s):  
Prime Ideal ◽  
Polynomial Ring ◽  
Commutative Algebra ◽  
Laurent Polynomial ◽  
Polynomial Rings ◽  
Prime Ideals ◽  
Ideal Structure ◽  
Locally Finite ◽  
Finite Dimensional ◽  
Laurent Polynomial Ring

Let R be a commutative ring and {σ1,…,σn} a set of commuting automorphisms of R. Let T = be the skew Laurent polynomial ring in n indeterminates over R and let be the Laurent polynomial ring in n central indeterminates over R. There is an isomorphism φ of right R-modules between T and S given by φ(θj) = xj. We will show that the map φ induces a bijection between the prime ideals of T and the Γ-prime ideals of S, where Γ is a certain set of endomorphisms of the ℤ-module S. We can study the structure of the lattice of Γ-prime ideals of the ring S by using commutative algebra, and this allows us to deduce results about the prime ideal structure of the ring T. As an example, if R is a Cohen-Macaulay ℂ-algebra and the action of the σj on R is locally finite-dimensional, we will show that the ring T is catenary.

scholarly journals Finite domination and Novikov rings: Laurent polynomial rings in two variables

2015 ◽  
Vol 14 (04) ◽  
pp. 1550055
Author(s):  
Thomas Hüttemann ◽  
David Quinn
Keyword(s):  
Polynomial Ring ◽  
Laurent Polynomial ◽  
Homotopy Equivalent ◽  
Polynomial Rings ◽  
Cochain Complex ◽  
Finitely Generated ◽  
Bounded Complex ◽  
Laurent Polynomial Ring ◽  
Free Modules

Let C be a bounded cochain complex of finitely generated free modules over the Laurent polynomial ring L = R[x, x-1, y, y-1]. The complex C is called R-finitely dominated if it is homotopy equivalent over R to a bounded complex of finitely generated projective R-modules. Our main result characterizes R-finitely dominated complexes in terms of Novikov cohomology: C is R-finitely dominated if and only if eight complexes derived from C are acyclic; these complexes are C ⊗L R〚x, y〛[(xy)-1] and C ⊗L R[x, x-1]〚y〛[y-1], and their variants obtained by swapping x and y, and replacing either indeterminate by its inverse.

scholarly journals Primitive skew Laurent polynomial rings

1978 ◽  
Vol 19 (1) ◽  
pp. 79-85 ◽  
Author(s):  
D. A. Jordan
Keyword(s):  
Polynomial Ring ◽  
Laurent Polynomial ◽  
Primitive Group ◽  
Polynomial Rings ◽  
Group Rings ◽  
Ore Extension ◽  
Laurent Polynomial Ring ◽  
Necessary And Sufficient

In [8] the author studied the question of the primitivity of an Ore extension R[x, δ], where δ is a derivation of the ring R. If a is an automorphism of R then it can be shown that R[x, α] is primitive if the following conditions are satisfied: (i) no power αsS ≥ 1, of α is inner; (ii) the only ideals of R invariant under α are 0 and R. These conditions are also known to be necessary and sufficient for the skew Laurent polynomial ring R[x, x−1, α] to be simple [9]. The object of this paper is to find conditions which are sufficient for R[x, x−1, α] to be primitive. The results obtained are remarkably similar to those of [8]. Two logically independent conditions are each found to be sufficient for the primitivity of R[x, x−1, α]. Of these, one is also shown to be sufficient for R[x, α] to be primitive. Included in the examples illustrating these results are some applications to the theory of primitive group rings. The basic techniques involved are also applied to produce a counterexample to the converse of a theorem of Goldie and Michler [3] on when R[x, x−1, α] is a Jacobson ring.

scholarly journals A Zariski--Nagata theorem for smooth ℤ-algebras

2020 ◽  
Vol 2020 (761) ◽  
pp. 123-140 ◽  
Author(s):  
Alessandro De Stefani ◽  
Eloísa Grifo ◽  
Jack Jeffries
Keyword(s):  
Prime Ideal ◽  
Polynomial Ring ◽  
Commutative Algebra ◽  
Classical Result ◽  
Differential Operators ◽  
Symbolic Power ◽  
Prime Ideals ◽  
Perfect Field ◽  
Mixed Characteristic ◽  
Symbolic Powers

AbstractIn a polynomial ring over a perfect field, the symbolic powers of a prime ideal can be described via differential operators: a classical result by Zariski and Nagata says that the n-th symbolic power of a given prime ideal consists of the elements that vanish up to order n on the corresponding variety. However, this description fails in mixed characteristic. In this paper, we use p-derivations, a notion due to Buium and Joyal, to define a new kind of differential powers in mixed characteristic, and prove that this new object does coincide with the symbolic powers of prime ideals. This seems to be the first application of p-derivations to commutative algebra.

scholarly journals Recent results on weakly factorial domains

2018 ◽  
Vol 20 ◽  
pp. 01001
Author(s):  
Chang Gyu Whan
Keyword(s):  
Polynomial Ring ◽  
Quotient Group ◽  
Integral Domain ◽  
Laurent Polynomial ◽  
Semigroup Ring ◽  
Prime Element ◽  
Cancellative Monoid ◽  
Factorial Domain ◽  
Laurent Polynomial Ring

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).

scholarly journals Multivariate Alexander colorings

2018 ◽  
Vol 27 (14) ◽  
pp. 1850076 ◽  
Author(s):  
Lorenzo Traldi
Keyword(s):  
Vector Space ◽  
Polynomial Ring ◽  
Knot Theory ◽  
Laurent Polynomial ◽  
Module Structure ◽  
Cyclic Covering ◽  
Covering Spaces ◽  
Linear Maps ◽  
Laurent Polynomial Ring ◽  
Special Case

We extend the notion of link colorings with values in an Alexander quandle to link colorings with values in a module [Formula: see text] over the Laurent polynomial ring [Formula: see text]. If [Formula: see text] is a diagram of a link [Formula: see text] with [Formula: see text] components, then the colorings of [Formula: see text] with values in [Formula: see text] form a [Formula: see text]-module [Formula: see text]. Extending a result of Inoue [Knot quandles and infinite cyclic covering spaces, Kodai Math. J. 33 (2010) 116–122], we show that [Formula: see text] is isomorphic to the module of [Formula: see text]-linear maps from the Alexander module of [Formula: see text] to [Formula: see text]. In particular, suppose [Formula: see text] is a field and [Formula: see text] is a homomorphism of rings with unity. Then [Formula: see text] defines a [Formula: see text]-module structure on [Formula: see text], which we denote [Formula: see text]. We show that the dimension of [Formula: see text] as a vector space over [Formula: see text] is determined by the images under [Formula: see text] of the elementary ideals of [Formula: see text]. This result applies in the special case of Fox tricolorings, which correspond to [Formula: see text] and [Formula: see text]. Examples show that even in this special case, the higher Alexander polynomials do not suffice to determine [Formula: see text]; this observation corrects erroneous statements of Inoue [Quandle homomorphisms of knot quandles to Alexander quandles, J. Knot Theory Ramifications 10 (2001) 813–821; op. cit.].

Restrictions on Homflypt and Kauffman polynomials arising from local moves

2020 ◽  
Vol 29 (06) ◽  
pp. 2050036
Author(s):  
Sandy Ganzell ◽  
Mercedes V. Gonzalez ◽  
Chloe’ Marcum ◽  
Nina Ryalls ◽  
Mariel Santos
Keyword(s):  
Polynomial Ring ◽  
Laurent Polynomial ◽  
Laurent Polynomial Ring

We study the effects of certain local moves on Homflyptand Kauffman polynomials. We show that all Homflypt(or Kauffman) polynomials are equal in a certain nontrivial quotient of the Laurent polynomial ring. As a consequence, we discover some new properties of these invariants.

On the Prime Ideals in a Commutative Ring

2000 ◽  
Vol 43 (3) ◽  
pp. 312-319 ◽  
Author(s):  
David E. Dobbs
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Polynomial Ring ◽  
Sufficient Conditions ◽  
Prime Ideals ◽  
Preceding Result ◽  
Finite Commutative Ring ◽  
The Axiom Of Choice ◽  
Necessary And Sufficient ◽  
Positive Integers

AbstractIf n and m are positive integers, necessary and sufficient conditions are given for the existence of a finite commutative ring R with exactly n elements and exactly m prime ideals. Next, assuming the Axiom of Choice, it is proved that if R is a commutative ring and T is a commutative R-algebra which is generated by a set I, then each chain of prime ideals of T lying over the same prime ideal of R has at most 2|I| elements. A polynomial ring example shows that the preceding result is best-possible.

scholarly journals Twisted Vertex Operators and Unitary Lie Algebras

2015 ◽  
Vol 67 (3) ◽  
pp. 573-596 ◽  
Author(s):  
Fulin Chen ◽  
Yun Gao ◽  
Naihuan Jing ◽  
Shaobin Tan
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Polynomial Ring ◽  
Central Extension ◽  
Laurent Polynomial ◽  
New Method ◽  
Vertex Operators ◽  
Irreducible Decomposition ◽  
Laurent Polynomial Ring

AbstractA representation of the central extension of the unitary Lie algebra coordinated with a skew Laurent polynomial ring is constructed using vertex operators over an integral ℤ2–lattice. The irreducible decomposition of the representation is explicitly computed and described. As a by–product, some fundamental representations of affine Kac–Moody Lie algebra of type A(2)n are recovered by the new method.

COLORINGS AND ALEXANDER POLYNOMIALS FOR RIBBON 2-KNOTS

2002 ◽  
Vol 11 (03) ◽  
pp. 403-412
Author(s):  
AKIKO SHIMA
Keyword(s):  
Polynomial Ring ◽  
Laurent Polynomial ◽  
Alexander Polynomials ◽  
Laurent Polynomial Ring

Let K be a ribbon 2-knot. We show that for any ideal J of the Laurent polynomial ring Λ, if Alexander polynomail of K is trivial, i.e., ΔK(t) = 1, then all colorings of K on Λ/J are trivial.

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