Jacobson radical of skew polynomial rings and skew group rings

1980 ◽  
Vol 35 (4) ◽  
pp. 327-338 ◽  
Author(s):  
Surinder Singh Bedi ◽  
Jai Ram
Keyword(s):  
Jacobson Radical ◽  
Polynomial Rings ◽  
Group Rings ◽  
Skew Polynomial Rings ◽  
Skew Group Rings ◽  
Skew Polynomial

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.

McCoy property and nilpotent elements of skew generalized power series rings

2017 ◽  
Vol 16 (10) ◽  
pp. 1750183 ◽  
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.

Partial Skew Polynomial Rings Over Semisimple Artinian Rings

2010 ◽  
Vol 38 (5) ◽  
pp. 1663-1676 ◽  
Author(s):  
Wagner Cortes ◽  
Miguel Ferrero ◽  
Yasuyuki Hirano ◽  
Hidetoshi Marubayashi
Keyword(s):  
Polynomial Rings ◽  
Artinian Rings ◽  
Skew Polynomial Rings ◽  
Skew Polynomial

ANNIHILATOR CONDITIONS IN MATRIX AND SKEW POLYNOMIAL RINGS

2012 ◽  
Vol 11 (04) ◽  
pp. 1250079 ◽  
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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