scholarly journals MAXIMAL COMMUTATIVE SUBRINGS AND SIMPLICITY OF ORE EXTENSIONS

2013 ◽  
Vol 12 (04) ◽  
pp. 1250192 ◽  
Author(s):  
JOHAN ÖINERT ◽  
JOHAN RICHTER ◽  
SERGEI D. SILVESTROV
Keyword(s):  
Polynomial Ring ◽  
Sufficient Conditions ◽  
Differential Polynomial ◽  
Nonzero Ideal ◽  
Polynomial Rings ◽  
Ore Extension ◽  
Differential Polynomial Ring ◽  
Ore Extensions ◽  
Necessary And Sufficient ◽  
Commutative Subring

The aim of this paper is to describe necessary and sufficient conditions for simplicity of Ore extension rings, with an emphasis on differential polynomial rings. We show that a differential polynomial ring, R[x; id R, δ], is simple if and only if its center is a field and R is δ-simple. When R is commutative we note that the centralizer of R in R[x; σ, δ] is a maximal commutative subring containing R and, in the case when σ = id R, we show that it intersects every nonzero ideal of R[x; id R, δ] nontrivially. Using this we show that if R is δ-simple and maximal commutative in R[x; id R, δ], then R[x; id R, δ] is simple. We also show that under some conditions on R the converse holds.

Simplicity of iterated Ore extensions

2021 ◽  
pp. 2250052
Author(s):  
Mamta Balodi ◽  
Sumit Kumar Upadhyay
Keyword(s):  
Polynomial Ring ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Differential Polynomial ◽  
Ore Extension ◽  
Differential Polynomial Ring ◽  
Ore Extensions ◽  
Commutative Domain ◽  
Necessary And Sufficient ◽  
Commutative Subring

Here we study the simplicity of an iterated Ore extension of a unital ring [Formula: see text]. We give necessary conditions for the simplicity of an iterated Ore extension when [Formula: see text] is a commutative domain. A class of iterated Ore extensions, namely the differential polynomial ring [Formula: see text] in [Formula: see text]-variables is considered. The conditions for a commutative domain [Formula: see text] of characteristic zero to be a maximal commutative subring of its differential polynomial ring [Formula: see text] are given, and the necessary and sufficient conditions for [Formula: see text] to be simple are also found.

Ore extensions of quasitriangular Hopf group coalgebras

2014 ◽  
Vol 13 (06) ◽  
pp. 1450016 ◽  
Author(s):  
Daowei Lu ◽  
Dingguo Wang
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Baxter Equation ◽  
Ore Extension ◽  
Ore Extensions ◽  
Necessary And Sufficient

In this paper, we mainly consider some special Ore extension of quasitriangular Hopf group coalgebra, and give the necessary and sufficient conditions when the Ore extension of quasitriangular Hopf group coalgebras will preserve the same quasitriangular structure. Furthermore, in the two examples given at the end, we construct new solutions of Yang–Baxter equation of Hopf group coalgebras version.

Jacobson Radicals of Skew Polynomial Rings of Derivation Type

2014 ◽  
Vol 57 (3) ◽  
pp. 609-613 ◽  
Author(s):  
Alireza Nasr-Isfahani
Keyword(s):  
Polynomial Ring ◽  
Jacobson Radical ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Polynomial Rings ◽  
Skew Polynomial Ring ◽  
Skew Polynomial Rings ◽  
Base Ring ◽  
Necessary And Sufficient ◽  
Skew Polynomial

AbstractWe provide necessary and sufficient conditions for a skew polynomial ring of derivation type to be semiprimitive when the base ring has no nonzero nil ideals. This extends existing results on the Jacobson radical of skew polynomial rings of derivation type.

Differential polynomial rings in several variables over locally nilpotent rings

2019 ◽  
Vol 30 (01) ◽  
pp. 117-123 ◽  
Author(s):  
Fei Yu Chen ◽  
Hannah Hagan ◽  
Allison Wang
Keyword(s):  
Polynomial Ring ◽  
Differential Polynomial ◽  
Polynomial Rings ◽  
Nilpotent Ring ◽  
Several Variables ◽  
Locally Nilpotent ◽  
Differential Polynomial Ring

We show that a differential polynomial ring over a locally nilpotent ring in several commuting variables is Behrens radical, extending a result by Chebotar.

scholarly journals ORE EXTENSIONS SATISFYING A POLYNOMIAL IDENTITY

2006 ◽  
Vol 05 (03) ◽  
pp. 287-306 ◽  
Author(s):  
ANDRÉ LEROY ◽  
JERZY MATCZUK
Keyword(s):  
Polynomial Identity ◽  
Semiprime Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Ore Extension ◽  
Ore Extensions ◽  
Necessary And Sufficient ◽  
Coefficient Ring ◽  
Injective Endomorphism

Necessary and sufficient conditions for an Ore extension S = R[x;σ,δ] to be a PI ring are given in the case σ is an injective endomorphism of a semiprime ring R satisfying the ACC on annihilators. Also, for an arbitrary endomorphism τ of R, a characterization of Ore extensions R[x;τ] which are PI rings is given, provided the coefficient ring R is noetherian.

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.

Derivations and bounded nilpotence index

2015 ◽  
Vol 25 (03) ◽  
pp. 433-438 ◽  
Author(s):  
Pace P. Nielsen ◽  
Michał Ziembowski
Keyword(s):  
Polynomial Ring ◽  
Differential Polynomial ◽  
Prime Radical ◽  
Polynomial Rings ◽  
Locally Nilpotent ◽  
Differential Polynomial Ring ◽  
Index Of Nilpotence ◽  
Nil Ring ◽  
Strong Barrier

We construct a nil ring R which has bounded index of nilpotence 2, is Wedderburn radical, and is commutative, and which also has a derivation δ for which the differential polynomial ring R[x;δ] is not even prime radical. This example gives a strong barrier to lifting certain radical properties from rings to differential polynomial rings. It also demarcates the strength of recent results about locally nilpotent PI rings.

Automorphisms and Derivations of Skew Polynomial Rings

1992 ◽  
Vol 35 (1) ◽  
pp. 126-132 ◽  
Author(s):  
Mary P. Rosen ◽  
Jerry D. Rosen
Keyword(s):  
Polynomial Ring ◽  
Prime Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Polynomial Rings ◽  
Skew Polynomial Ring ◽  
Skew Polynomial Rings ◽  
Inner Automorphisms ◽  
Necessary And Sufficient ◽  
Skew Polynomial

AbstractFor a prime ring R and σ ∊ Aut(R), we determine the group of Rstabilizing automorphisms of the skew polynomial ring R[x; σ]. In the case where R is simple, we characterize the X-inner automorphisms of R[x; σ]. We also provide necessary and sufficient conditions for a σ -commuting derivation of a prime ring R to extend to a derivation of R[x; σ].

scholarly journals ON PRIME-LIKE RADICALS

2010 ◽  
Vol 82 (1) ◽  
pp. 113-119 ◽  
Author(s):  
S. TUMURBAT ◽  
H. FRANCE-JACKSON
Keyword(s):  
Polynomial Ring ◽  
Prime Ring ◽  
Open Problem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Prime Radical ◽  
Polynomial Rings ◽  
Necessary And Sufficient

AbstractA radical γ is prime-like if, for every prime ring A, the polynomial ring A[x] is γ-semisimple. In this paper, we study properties of prime-like radicals. In particular, we give necessary and sufficient conditions for a radical γ containing the prime radical β to be prime-like. This allows us to easily find distinct special radicals that coincide on simple rings and on polynomial rings, which answers a question put by Ferrero. It also allows us to reformulate a long-standing open problem of Gardner in terms of prime-like radicals.

ON THE CONNECTION BETWEEN DIFFERENTIAL POLYNOMIAL RINGS AND POLYNOMIAL RINGS OVER NIL RINGS

2019 ◽  
Vol 101 (3) ◽  
pp. 438-441
Author(s):  
LOUISA CATALANO ◽  
MEGAN CHANG-LEE
Keyword(s):  
Polynomial Ring ◽  
Positive Characteristic ◽  
Differential Polynomial ◽  
Polynomial Rings ◽  
Locally Nilpotent Derivation ◽  
Locally Nilpotent ◽  
Nilpotent Derivation ◽  
Differential Polynomial Ring ◽  
Field Of Positive Characteristic ◽  
Algebra Over A Field

In this paper, we study some connections between the polynomial ring $R[y]$ and the differential polynomial ring $R[x;D]$. In particular, we answer a question posed by Smoktunowicz, which asks whether $R[y]$ is nil when $R[x;D]$ is nil, provided that $R$ is an algebra over a field of positive characteristic and $D$ is a locally nilpotent derivation.

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