Sagbi bases of Cox–Nagata rings

Bernd Sturmfels; Zhiqiang Xu

Displaying similar documents to “Sagbi bases of Cox–Nagata rings”

A Characterization of One-Element p-Bases of Rings of Constants

Piotr Jędrzejewicz (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let K be a unique factorization domain of characteristic p > 0, and let f ∈ K[x₁,...,xₙ] be a polynomial not lying in $K [x ₁^{p}, . . ., x ₙ^{p}]$ . We prove that $K [x ₁^{p}, . . ., x ₙ^{p}, f]$ is the ring of constants of a K-derivation of K[x₁,...,xₙ] if and only if all the partial derivatives of f are relatively prime. The proof is based on a generalization of Freudenburg’s lemma to the case of polynomials over a unique factorization domain of arbitrary characteristic.

Generalized E-algebras via λ-calculus I

Rüdiger Göbel, Saharon Shelah (2006)

Fundamenta Mathematicae

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An R-algebra A is called an E(R)-algebra if the canonical homomorphism from A to the endomorphism algebra $E n d_{R} A$ of the R-module $_{R} A$ , taking any a ∈ A to the right multiplication $a_{r} \in E n d_{R} A$ by a, is an isomorphism of algebras. In this case $_{R} A$ is called an E(R)-module. There is a proper class of examples constructed in [4]. E(R)-algebras arise naturally in various topics of algebra. So it is not surprising that they were investigated thoroughly in the last decades; see [3, 5, 7, 8, 10, 13, 14, 15, 18,...

Cartan-Eilenberg projective, injective and flat complexes

Xiaorui Zhai, Chunxia Zhang (2016)

Czechoslovak Mathematical Journal

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Let $R$ be an associative ring with identity and $ℱ$ a class of $R$ -modules. In this article: we first give a detailed treatment of Cartan-Eilenberg $ℱ$ complexes and extend the basic properties of the class $ℱ$ to the class $CE (ℱ$ ). Secondly, we study and give some equivalent characterizations of Cartan-Eilenberg projective, injective and flat complexes which are similar to projective, injective and flat modules, respectively. As applications, we characterize some classical rings in terms of these...

About G-rings

Najib Mahdou (2017)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we are concerned with G-rings. We generalize the Kaplansky’s theorem to rings with zero-divisors. Also, we assert that if $R \subseteq T$ is a ring extension such that $m T \subseteq R$ for some regular element $m$ of $T$ , then $T$ is a G-ring if and only if so is $R$ . Also, we examine the transfer of the G-ring property to trivial ring extensions. Finally, we conclude the paper with illustrative examples discussing the utility and limits of our results.

Rings of $(γ, δ)$ type

Kleinfeld, Erwin (1959)

Portugaliae mathematica

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Bounds for quotients in rings of formal power series with growth constraints

Vincent Thilliez (2002)

Studia Mathematica

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In rings $Γ_{M}$ of formal power series in several variables whose growth of coefficients is controlled by a suitable sequence $M = {(M_{l})}_{l \geq 0}$ (such as rings of Gevrey series), we find precise estimates for quotients F/Φ, where F and Φ are series in $Γ_{M}$ such that F is divisible by Φ in the usual ring of all power series. We give first a simple proof of the fact that F/Φ belongs also to $Γ_{M}$ , provided $Γ_{M}$ is stable under derivation. By a further development of the method, we obtain the main result of the paper,...

A commutativity theorem for associative rings

Mohammad Ashraf (1995)

Archivum Mathematicum

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Let $m > 1, s \geq 1$ be fixed positive integers, and let $R$ be a ring with unity $1$ in which for every $x$ in $R$ there exist integers $p = p (x) \geq 0, q = q (x) \geq 0, n = n (x) \geq 0, r = r (x) \geq 0$ such that either $x^{p} [x^{n}, y] x^{q} = x^{r} [x, y^{m}] y^{s}$ or $x^{p} [x^{n}, y] x^{q} = y^{s} [x, y^{m}] x^{r}$ for all $y \in R$ . In the present paper it is shown that $R$ is commutative if it satisfies the property $Q (m)$ (i.e. for all $x, y \in R, m [x, y] = 0$ implies $[x, y] = 0$ ).

A unified approach to the Armendariz property of polynomial rings and power series rings

Tsiu-Kwen Lee, Yiqiang Zhou (2008)

Colloquium Mathematicae

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A ring R is called Armendariz (resp., Armendariz of power series type) if, whenever $(\sum_{i \geq 0} a_{i} x^{i}) (\sum_{j \geq 0} b_{j} x^{j}) = 0$ in R[x] (resp., in R[[x]]), then $a_{i} b_{j} = 0$ for all i and j. This paper deals with a unified generalization of the two concepts (see Definition 2). Some known results on Armendariz rings are extended to this more general situation and new results are obtained as consequences. For instance, it is proved that a ring R is Armendariz of power series type iff the same is true of R[[x]]. For an injective endomorphism...

Vortex rings for the Gross-Pitaevskii equation

Fabrice Bethuel, G. Orlandi, Didier Smets (2004)

Journal of the European Mathematical Society

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We provide a mathematical proof of the existence of traveling vortex rings solutions to the Gross–Pitaevskii (GP) equation in dimension $N \geq 3$ . We also extend the asymptotic analysis of the free field Ginzburg–Landau equation to a larger class of equations, including the Ginzburg–Landau equation for superconductivity as well as the traveling wave equation for GP. In particular we rigorously derive a curvature equation for the concentration set (i.e. line vortices if $N = 3$ ).

On the Gram-Schmidt orthonormalizatons of subsystems of Schauder systems

Robert E. Zink (2002)

Colloquium Mathematicae

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In one of the earliest monographs that involve the notion of a Schauder basis, Franklin showed that the Gram-Schmidt orthonormalization of a certain Schauder basis for the Banach space of functions continuous on [0,1] is again a Schauder basis for that space. Subsequently, Ciesielski observed that the Gram-Schmidt orthonormalization of any Schauder system is a Schauder basis not only for C[0,1], but also for each of the spaces $L^{p} [0, 1]$ , 1 ≤ p < ∞. Although perhaps not probable, the latter...

On matrix Lie rings over a commutative ring that contain the special linear Lie ring

Evgenii L. Bashkirov, Esra Pekönür (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let $K$ be an associative and commutative ring with $1$ , $k$ a subring of $K$ such that $1 \in k$ , $n \geq 2$ an integer. The paper describes subrings of the general linear Lie ring $g l_{n} (K)$ that contain the Lie ring of all traceless matrices over $k$ .

Twisted group rings of strongly unbounded representation type

Leonid F. Barannyk, Dariusz Klein (2004)

Colloquium Mathematicae

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Let S be a commutative local ring of characteristic p, which is not a field, S* the multiplicative group of S, W a subgroup of S*, G a finite p-group, and $S^{λ} G$ a twisted group ring of the group G and of the ring S with a 2-cocycle λ ∈ Z²(G,S*). Denote by $I n d_{m} (S^{λ} G)$ the set of isomorphism classes of indecomposable $S^{λ} G$ -modules of S-rank m. We exhibit rings $S^{λ} G$ for which there exists a function $f_{λ} : ℕ \to ℕ$ such that $f_{λ} (n) \geq n$ and $I n d_{f_{λ} (n)} (S^{λ} G)$ is an infinite set for every natural n > 1. In special cases $f_{λ} (ℕ)$ contains every natural...

Necessary condition for Kostyuchenko type systems to be a basis in Lebesgue spaces

Aydin Sh. Shukurov (2012)

Colloquium Mathematicae

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A necessary condition for Kostyuchenko type systems and system of powers to be a basis in $L_{p}$ (1 ≤ p < +∞) spaces is obtained. In particular, we find a necessary condition for a Kostyuchenko system to be a basis in $L_{p}$ (1 ≤ p < +∞).

A subclass of strongly clean rings

Orhan Gurgun, Sait Halicioglu and Burcu Ungor (2015)

Communications in Mathematics

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In this paper, we introduce a subclass of strongly clean rings. Let $R$ be a ring with identity, $J$ be the Jacobson radical of $R$ , and let $J^{#}$ denote the set of all elements of $R$ which are nilpotent in $R / J$ . An element $a \in R$ is called provided that there exists an idempotent $e \in R$ such that $a e = e a$ and $a - e$ or $a + e$ is an element of $J^{#}$ . A ring $R$ is said to be in case every element in $R$ is very $J^{#}$ -clean. We prove that every very $J^{#}$ -clean ring is strongly $π$ -rad clean and has stable range one. It is shown that for a...

Torsion points on families of simple abelian surfaces and Pell's equation over polynomial rings (with an appendix by E. V. Flynn)

David Masser, Umberto Zannier (2015)

Journal of the European Mathematical Society

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In recent papers we proved a special case of a variant of Pink’s Conjecture for a variety inside a semiabelian scheme: namely for any curve inside anything isogenous to a product of two elliptic schemes. Here we go beyond the elliptic situation by settling the crucial case of any simple abelian surface scheme defined over the field of algebraic numbers, thus confirming an earlier conjecture of Shou-Wu Zhang. This is of particular relevance in the topic, also in view of very recent counterexamples...

Skeletons, bodies and generalized $E (R)$ -algebras

Rüdiger Göbel, Daniel Herden, Saharon Shelah (2009)

Journal of the European Mathematical Society

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On feebly nil-clean rings

Marjan Sheibani Abdolyousefi, Neda Pouyan (2024)

Czechoslovak Mathematical Journal

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A ring $R$ is feebly nil-clean if for any $a \in R$ there exist two orthogonal idempotents $e, f \in R$ and a nilpotent $w \in R$ such that $a = e - f + w$ . Let $R$ be a 2-primal feebly nil-clean ring. We prove that every matrix ring over $R$ is feebly nil-clean. The result for rings of bounded index is also obtained. These provide many classes of rings over which every matrix is the sum of orthogonal idempotent and nilpotent matrices.

The rings which are Boolean

Ivan Chajda, Filip Švrček (2011)

Discussiones Mathematicae - General Algebra and Applications

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We study unitary rings of characteristic 2 satisfying identity $x^{p} = x$ for some natural number p. We characterize several infinite families of these rings which are Boolean, i.e., every element is idempotent. For example, it is in the case if $p = 2^{n} - 2$ or $p = 2^{n} - 5$ or $p = 2^{n} + 1$ for a suitable natural number n. Some other (more general) cases are solved for p expressed in the form $2^{q} + 2 m + 1$ or $2^{q} + 2 m$ where q is a natural number and $m \in 1, 2, . . ., 2^{q} - 1$ .