On Radicals of Skew Inverse Laurent Series Rings

2016 ◽  
Vol 23 (02) ◽  
pp. 335-346
Author(s):  
A. Moussavi
Keyword(s):  
Polynomial Ring ◽  
Jacobson Radical ◽  
Laurent Series ◽  
Ring Extensions ◽  
Series Type ◽  
Armendariz Ring

Let R be a ring and α an automorphism of R. Amitsur proved that the Jacobson radical J(R[x]) of the polynomial ring R[x] is the polynomial ring over the nil ideal J(R[x]) ∩ R. Following Amitsur, it is shown that when R is an Armendariz ring of skew inverse Laurent series type and S is any one of the ring extensions R[x;α], R[x,x-1;α], R[[x-1;α]] and R((x-1;α)), then ℜ𝔞𝔡(S) = ℜ𝔞𝔡(R)S = Nil (S), ℜ𝔞𝔡(S) ∩ R = Nil (R), where ℜ𝔞𝔡 is a radical in a class of radicals which includes the Wedderburn, lower nil, Levitzky and upper nil radicals.

RINGS OVER WHICH COEFFICIENTS OF NILPOTENT POLYNOMIALS ARE NILPOTENT

2011 ◽  
Vol 21 (05) ◽  
pp. 745-762 ◽  
Author(s):  
TAI KEUN KWAK ◽  
YANG LEE
Keyword(s):  
Polynomial Ring ◽  
Regular Ring ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Nilpotent Elements ◽  
Von Neumann Regular ◽  
Armendariz Rings ◽  
Ring Extensions ◽  
Nil Ring

Antoine studied conditions which are connected to the question of Amitsur of whether or not a polynomial ring over a nil ring is nil, observing the structure of nilpotent elements in Armendariz rings and introducing the notion of nil-Armendariz rings. The class of nil-Armendariz rings contains Armendariz rings and NI rings. We continue the study of nil-Armendariz rings, concentrating on the structure of rings over which coefficients of nilpotent polynomials are nilpotent. In the procedure we introduce the notion of CN-rings that is a generalization of nil-Armendariz rings. We first construct a CN-ring but not nil-Armendariz. This may be a base on which Antoine's theory can be applied and elaborated. We investigate basic ring theoretic properties of CN-rings, and observe various kinds of CN-rings including ordinary ring extensions. It is shown that a ring R is CN if and only if R is nil-Armendariz if and only if R is Armendariz if and only if R is reduced when R is a von Neumann regular ring.

On Skew Armendariz of Laurent Series Type Rings

2012 ◽  
Vol 40 (11) ◽  
pp. 3999-4018 ◽  
Author(s):  
M. Habibi ◽  
A. Moussavi ◽  
S. Mokhtari
Keyword(s):  
Laurent Series ◽  
Series Type

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.

On α-nilpotent elements and α-Armendariz rings

2015 ◽  
Vol 14 (05) ◽  
pp. 1550064
Author(s):  
Hong Kee Kim ◽  
Nam Kyun Kim ◽  
Tai Keun Kwak ◽  
Yang Lee ◽  
Hidetoshi Marubayashi
Keyword(s):  
Polynomial Ring ◽  
Polynomial Rings ◽  
Skew Polynomial Ring ◽  
Nilpotent Elements ◽  
Skew Polynomial Rings ◽  
Armendariz Rings ◽  
Skew Polynomial ◽  
Armendariz Ring

Antoine studied the structure of the set of nilpotent elements in Armendariz rings and introduced the concept of nil-Armendariz property as a generalization. Hong et al. studied Armendariz property on skew polynomial rings and introduced the notion of an α-Armendariz ring, where α is a ring monomorphism. In this paper, we investigate the structure of the set of α-nilpotent elements in α-Armendariz rings and introduce an α-nil-Armendariz ring. We examine the set of [Formula: see text]-nilpotent elements in a skew polynomial ring R[x;α], where [Formula: see text] is the monomorphism induced by the monomorphism α of an α-Armendariz ring R. We prove that every polynomial with α-nilpotent coefficients in a ring R is [Formula: see text]-nilpotent when R is of bounded index of α-nilpotency, and moreover, R is shown to be α-nil-Armendariz in this situation. We also characterize the structure of the set of α-nilpotent elements in α-nil-Armendariz rings, and investigate the relations between α-(nil-)Armendariz property and other standard ring theoretic properties.

scholarly journals Radicals Of Polynomial Rings

1956 ◽  
Vol 8 ◽  
pp. 355-361 ◽  
Author(s):  
S. A. Amitsur
Keyword(s):  
Polynomial Ring ◽  
Jacobson Radical ◽  
Present Note ◽  
Polynomial Rings ◽  
Lower Radical

Introduction. Let R be a ring and let R[x] be the ring of all polynomials in a commutative indeterminate x over R. Let J(R) denote the Jacobson radical (5) of the ring R and let L(R) be the lower radical (4) of R. The main object of the present note is to determine the radicals J(R[x]) and L(R[x]). The Jacobson radical J(R[x]) is shown to be a polynomial ring N[x] over a nil ideal N of R and the lower radical L(R[x]) is the polynomial ring L(R)[x].

ON Π-NEAR-ARMENDARIZ RINGS

2009 ◽  
Vol 02 (01) ◽  
pp. 77-83
Author(s):  
Sh. Ghalandarzadeh ◽  
P. Malakooti Rad
Keyword(s):  
Basic Properties ◽  
Armendariz Rings ◽  
Ring Extensions ◽  
Armendariz Ring

In this note we introduce a concept, so-called π-Near-Armendariz ring, that is a generalization of both Armendariz rings and 2-primal rings. We first observe the basic properties of π-Near-Armendariz rings, constructing typical examples. We next extend the class of π-Near-Armendariz rings, through various ring extensions.

Extensions of Generalized Reduced Rings

2005 ◽  
Vol 12 (02) ◽  
pp. 229-240 ◽  
Author(s):  
Chan Yong Hong ◽  
Nam Kyun Kim ◽  
Tai Keun Kwak
Keyword(s):  
Polynomial Ring ◽  
Armendariz Ring

Anderson and Camillo studied the class of rings satisfying ZCn for n ≥ 2, which is a generalization of reduced rings. In this paper, we continue the study of such rings. We observe several extensions of rings satisfying ZCn. Rings satisfying the zero insertion property for n (simply, ZIn), which is a generalization of ZCn, are also introduced. In particular, we prove that every ring satisfying ZIn for some n ≥ 2 is a 2-primal ring. Furthermore, if R is an Armendariz ring satisfying ZIn for n ≥ 2, then the polynomial ring R[x] over R also satisfies ZIn.

On nilpotent elements of ore extensions

2017 ◽  
Vol 10 (03) ◽  
pp. 1750043
Author(s):  
Masoud Azimi ◽  
Ahmad Moussavi
Keyword(s):  
Power Series ◽  
Associative Ring ◽  
General Setting ◽  
Nilpotent Elements ◽  
Ore Extensions ◽  
Ring Extensions ◽  
Series Type

Let [Formula: see text] be an associative ring with unity, [Formula: see text] be an endomorphism of [Formula: see text] and [Formula: see text] an [Formula: see text]-derivation of [Formula: see text]. We introduce the notion of [Formula: see text]-nilpotent p.p.-rings, and prove that the [Formula: see text]-nilpotent p.p.-condition extends to various ring extensions. Among other results, we show that, when [Formula: see text] is a nil-[Formula: see text]-compatible and [Formula: see text]-primal ring with [Formula: see text] nilpotent, then [Formula: see text]; and when [Formula: see text] is a nil Armendriz ring of skew power series type with [Formula: see text] nilpotent, then [Formula: see text] where [Formula: see text] is the set of nilpotent elements of [Formula: see text]. These results extend existing results to a more general setting.

A polynomial ring that is Jacobson radical and not nil

2001 ◽  
Vol 124 (1) ◽  
pp. 317-325 ◽  
Author(s):  
Agata Smoktunowicz ◽  
E. R. Puczyłowski
Keyword(s):  
Polynomial Ring ◽  
Jacobson Radical
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