Quantum Drinfeld Hecke Algebras

AbstractWe consider finite groups acting on quantum (or skew) polynomial rings. Deformations of the semidirect product of the quantum polynomial ring with the acting group extend symplectic reflection algebras and graded Hecke algebras to the quantum setting over a field of arbitrary characteristic. We give necessary and sufficient conditions for such algebras to satisfy a Poincaré–Birkhoff–Witt property using the theory of noncommutative Gröbner bases. We include applications to the case of abelian groups and the case of groups acting on coordinate rings of quantum planes. In addition, we classify graded automorphisms of the coordinate ring of quantum 3-space. In characteristic zero, Hochschild cohomology gives an elegant description of the Poincaré–Birkhoff–Witt conditions.

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Jacobson Radicals of Skew Polynomial Rings of Derivation Type

Canadian Mathematical Bulletin ◽

10.4153/cmb-2014-018-9 ◽

2014 ◽

Vol 57 (3) ◽

pp. 609-613 ◽

Cited By ~ 3

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.

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Automorphisms and Derivations of Skew Polynomial Rings

Canadian Mathematical Bulletin ◽

10.4153/cmb-1992-018-6 ◽

1992 ◽

Vol 35 (1) ◽

pp. 126-132 ◽

Cited By ~ 3

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].

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ANNIHILATOR CONDITIONS IN MATRIX AND SKEW POLYNOMIAL RINGS

Journal of Algebra and Its Applications ◽

10.1142/s021949881250079x ◽

2012 ◽

Vol 11 (04) ◽

pp. 1250079 ◽

Cited By ~ 7

Author(s):

A. ALHEVAZ ◽

A. MOUSSAVI

Keyword(s):

Polynomial Rings ◽

Trivial Extension ◽

Skew Polynomial Ring ◽

Matrix Rings ◽

Skew Polynomial Rings ◽

Armendariz Rings ◽

Ore Extensions ◽

Necessary And Sufficient ◽

Skew Polynomial ◽

Armendariz Ring

Let R be a ring with an endomorphism α and α-derivation δ. By [A. R. Nasr-Isfahani and A. Moussavi, Ore extensions of skew Armendariz rings, Comm. Algebra 36(2) (2008) 508–522], a ring R is called a skew Armendariz ring, if for polynomials f(x) = a0 + a1 x + ⋯ + anxn, g(x) = b0+b1x + ⋯ + bmxm in R[x; α, δ], f(x)g(x) = 0 implies a0bj = 0 for each 0 ≤ j ≤ m. In this paper, radicals of the skew polynomial ring R[x; α, δ], in terms of a skew Armendariz ring R, is determined. We prove that several properties transfer between R and R[x; α, δ], in case R is an α-compatible skew Armendariz ring. We also identify some "relatively maximal" skew Armendariz subrings of matrix rings, and obtain a necessary and sufficient condition for a trivial extension to be skew Armendariz. Consequently, new families of non-reduced skew Armendariz rings are presented and several known results related to Armendariz rings and skew polynomial rings will be extended and unified.

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A GENERALIZATION OF NIL-ARMENDARIZ RINGS

Journal of Algebra and Its Applications ◽

10.1142/s0219498813500011 ◽

2013 ◽

Vol 12 (06) ◽

pp. 1350001 ◽

Cited By ~ 1

Author(s):

MOHAMMAD HABIBI ◽

RAOUFEH MANAVIYAT

Keyword(s):

Sufficient Conditions ◽

Laurent Polynomial ◽

Polynomial Rings ◽

Monoid Ring ◽

Skew Polynomial Rings ◽

Monoid Homomorphism ◽

Special Cases ◽

Armendariz Rings ◽

Skew Polynomial ◽

The Relationship

Let R be a ring, M a monoid and ω : M → End (R) a monoid homomorphism. The skew monoid ring R * M is a common generalization of polynomial rings, skew polynomial rings, (skew) Laurent polynomial rings and monoid rings. In the current work, we study the nil skew M-Armendariz condition on R, a generalization of the standard nil-Armendariz condition from polynomials to skew monoid rings. We resolve the structure of nil skew M-Armendariz rings and obtain various necessary or sufficient conditions for a ring to be nil skew M-Armendariz, unifying and generalizing a number of known nil Armendariz-like conditions in the aforementioned special cases. We consider central idempotents which are invariant under a monoid endomorphism of nil skew M-Armendariz rings and classify how the nil skew M-Armendariz rings behaves under various ring extensions. We also provide rich classes of skew monoid rings which satisfy in a condition nil (R * M) = nil (R) * M. Moreover, we study on the relationship between the zip and weak zip properties of a ring R and those of the skew monoid ring R * M.

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Nilpotent Elements and Skew Polynomial Rings

Algebra Colloquium ◽

10.1142/s1005386712000715 ◽

2012 ◽

Vol 19 (spec01) ◽

pp. 821-840 ◽

Cited By ~ 2

Author(s):

A. Alhevaz ◽

A. Moussavi ◽

E. Hashemi

Keyword(s):

Polynomial Ring ◽

Sufficient Conditions ◽

Polynomial Rings ◽

Skew Polynomial Ring ◽

Nilpotent Elements ◽

Skew Polynomial Rings ◽

Special Cases ◽

Ring Extensions ◽

Skew Polynomial

We study the structure of the set of nilpotent elements in extended semicommutative rings and introduce nil α-semicommutative rings as a generalization. We resolve the structure of nil α-semicommutative rings and obtain various necessary or sufficient conditions for a ring to be nil α-semicommutative, unifying and generalizing a number of known commutative-like conditions in special cases. We also classify which of the standard nilpotence properties on polynomial rings pass to skew polynomial ring. Constructing various examples, we classify how the nil α-semicommutative rings behaves under various ring extensions. Also, we consider the nil-Armendariz condition on a skew polynomial ring.

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McCoy property and nilpotent elements of skew generalized power series rings

Journal of Algebra and Its Applications ◽

10.1142/s0219498817501833 ◽

2017 ◽

Vol 16 (10) ◽

pp. 1750183 ◽

Cited By ~ 2

Author(s):

Kamal Paykan ◽

Ahmad Moussavi

Keyword(s):

Power Series ◽

Sufficient Conditions ◽

Polynomial Rings ◽

Group Rings ◽

Generalized Power Series ◽

Series Ring ◽

Skew Polynomial Rings ◽

Monoid Homomorphism ◽

Power Series Rings ◽

Skew Polynomial

Let [Formula: see text] be a ring, [Formula: see text] a strictly ordered monoid and [Formula: see text] a monoid homomorphism. The skew generalized power series ring [Formula: see text] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev–Neumann Laurent series rings. In this paper, we consider the problem of determining when [Formula: see text] is nilpotent in [Formula: see text]. We study various annihilator properties and a variety of conditions and related properties that the skew generalized power series [Formula: see text] inherits from [Formula: see text]. We also introduce and study the [Formula: see text]-McCoy condition on [Formula: see text], a generalization of the standard McCoy condition from polynomials to skew generalized power series. We resolve the structure of [Formula: see text]-McCoy rings and obtain various necessary or sufficient conditions for a ring to be [Formula: see text]-McCoy. As particular cases of our general results we obtain several new theorems on the McCoy condition. Moreover various examples of [Formula: see text]-McCoy rings are provided.

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