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Polynomial Rings over Pseudovaluation Rings [PDF]
International Journal of Mathematics and Mathematical Sciences, 2007 Let R be a ring. Let σ be an automorphism of R. We define a σ-divided ring and prove the following. (1) Let R be a commutative pseudovaluation ring such that x∉P for any P∈Spec(R[x,σ]) . Then R[x,σ] is also a pseudovaluation ring.
V. K. Bhat
doaj +3 more sources
On the Waring problem for polynomial rings. [PDF]
Proc Natl Acad Sci U S A, 2012 In this note we discuss an analog of the classical Waring problem for . Namely, we show that a general homogeneous polynomial of degree divisible by k≥2 can be represented as a sum of at most kn k-th powers of homogeneous polynomials in .
Fröberg R, Ottaviani G, Shapiro B.
europepmc +3 more sources
Differential polynomial rings over rings satisfying a polynomial identity [PDF]
Journal of Algebra, 2014 Let $R$ be a ring satisfying a polynomial identity and let $ $ be a derivation of $R$. We show that if $N$ is the nil radical of $R$ then $ (N)\subseteq N$ and the Jacobson radical of $R[x; ]$ is equal to $N[x; ]$. As a consequence, we have that if $R$ is locally nilpotent then $R[x; ]$ is locally nilpotent.
J. Bell, Blake Madill, Forte Shinko
semanticscholar +4 more sources
Ring-LWE in Polynomial Rings [PDF]
IACR Cryptology ePrint Archive, 2012 The Ring-LWE problem, introduced by Lyubashevsky, Peikert, and Regev (Eurocrypt 2010), has been steadily finding many uses in numerous cryptographic applications. Still, the Ring-LWE problem defined in [LPR10] involves the fractional ideal R ∨, the dual of the ring R , which is the source of many theoretical and implementation technicalities. Until now,
L. Ducas, A. Durmus
semanticscholar +3 more sources
Semifields from skew polynomial rings [PDF]
, 2012 Skew polynomial rings were used to construct finite semifields by Petit in 1966, following from a construction of Ore and Jacobson of associative division algebras.
Lavrauw, Michel, Sheekey, John
core +6 more sources
Simple ambiskew polynomial rings [PDF]
Journal of Algebra, 2013 We determine simplicity criteria in characteristics 0 and $p$ for a ubiquitous class of iterated skew polynomial rings in two indeterminates over a base ring. One obstruction to simplicity is the possible existence of a canonical normal element $z$. In the case where this element exists we give simplicity criteria for the rings obtained by inverting $z$
D. Jordan, Imogen E. Wells
semanticscholar +3 more sources
The Regularity Conjecture for prime ideals in polynomial rings
EMS Surveys in Mathematical Sciences, 2021 This paper presents a survey on recent developments on regularity of prime ideals in polynomial rings.
J. McCullough, I. Peeva
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Nilpotent graphs of skew polynomial rings over non-commutative rings [PDF]
Transactions on Combinatorics, 2020 Let $R$ be a ring and $\alpha$ be a ring endomorphism of $R$. The undirected nilpotent graph of $R$, denoted by $\Gamma_N(R)$, is a graph with vertex set $Z_N(R)^*$, and two distinct vertices $x$ and $y$ are connected by an edge if and only if ...
Mohammad Javad Nikmehr, Abdolreza Azadi
doaj +1 more source
Big polynomial rings and Stillman’s conjecture [PDF]
Inventiones Mathematicae, 2018 Ananyan-Hochster's recent proof of Stillman's conjecture reveals a key principle: if $f_1, \dots, f_r$ are elements of a polynomial ring such that no linear combination has small strength then $f_1, \dots, f_r$ behave approximately like independent ...
D. Erman, Steven V. Sam, Andrew Snowden
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Proceedings of the American Mathematical Society, 1970 For aii algebra R over a field k, with residue field K to be a ring of polyniomials in one variable over k it is necessary that trdeg K/k = 1. We prove that under the hypothesis tr* deg K/k -1, R is a ring of Krtull-dimension at most one. This is used to derive sufficient conditions for R to be a ring of polynomials in one variable over k. 1.
Evyatar, A., Zaks, A.
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