Near-Ring Radicals and Class Pairs

2005 ◽  
Vol 12 (01) ◽  
pp. 101-112
Author(s):  
L. Godloza ◽  
N. J. Groenewald ◽  
W. A. Olivier
Keyword(s):  
General Class ◽  
Prime Radical ◽  
Radical Class ◽  
Radical Pairs ◽  
Semisimple Class ◽  
The Relationship ◽  
Class Pair

For near-ring ideal mappings ρ1 and ρ2, we investigate radical theoretical properties of and the relationship among the class pairs (ρ1: ρ2), [Formula: see text] and (ℛρ2: ℛρ1). Conditions on ρ1 and ρ2 are given for a general class pair to form a radical class of various types. These types include the Plotkin and KA-radical varieties. A number of examples are shown to motivate the suitability of the theory of Hoehnke-radicals over KA-radicals when radical pairs of near-rings are studied. In particular, it is shown that [Formula: see text] forms a KA-radical class, where [Formula: see text] denotes the class of completely prime near-rings and [Formula: see text] the class of 3-prime near-rings. This gives another near-ring generalization of the 2-primal ring concept. The theory of radical pairs are also used to show that in general the class of 3-semiprime near-rings is not the semisimple class of the 3-prime radical.

On Jacobson Near-rings and Special Radicals

2007 ◽  
Vol 14 (01) ◽  
pp. 1-14
Author(s):  
L. Godloza ◽  
N. J. Groenewald ◽  
W. A. Olivier
Keyword(s):  
Necessary Conditions ◽  
Radical Class ◽  
Special Radical ◽  
The Relationship ◽  
Class Pair

In this paper, we construct special radicals using class pairs of near-rings. We establish necessary conditions for a class pair to be a special radical class. We then define Jacobson-type near-rings and show that in most cases the class of all near-rings of this type is a special radical class. Subsequently, we investigate the relationship between each Jacobson-type near-ring and the corresponding matrix near-ring.

scholarly journals On the closure of the prime radical of a Banach algebra

1993 ◽  
Vol 36 (3) ◽  
pp. 421-425 ◽  
Author(s):  
D. W. B. Somerset ◽  
G. A. Willis
Keyword(s):  
Banach Algebra ◽  
Prime Radical ◽  
Prime Ideals ◽  
The Relationship

The relationship between the prime ideals and the primal ideals of a Banach algebra is investigated. It is shown that the closure of the prime radical of a Banach algebra may be properly contained in the intersection of the closed primal ideals of the algebra.

scholarly journals ON SUPERNILPOTENT RADICALS WITH THE AMITSUR PROPERTY

2009 ◽  
Vol 80 (3) ◽  
pp. 423-429 ◽  
Author(s):  
HALINA FRANCE-JACKSON
Keyword(s):  
Jacobson Radical ◽  
Semiprime Ring ◽  
Nonzero Ideal ◽  
Prime Radical ◽  
Factor Ring ◽  
Semisimple Class ◽  
Primitive Ring

AbstractA radical α has the Amitsur property if α(A[x])=(α(A[x])∩A)[x] for all rings A. For rings R⊆S with the same unity, we call S a finite centralizing extension of R if there exist b1,b2,…,bt∈S such that S=b1R+b2R+⋯+btR and bir=rbi for all r∈R and i=1,2,…,t. A radical α is FCE-friendly if α(S)∩R⊆α(R) for any finite centralizing extension S of a ring R. We show that if α is a supernilpotent radical whose semisimple class contains the ring ℤ of all integers and α is FCE-friendly, then α has the Amitsur property. In this way the Amitsur property of many well-known radicals such as the prime radical, the Jacobson radical, the Brown–McCoy radical, the antisimple radical and the Behrens radical can be established. Moreover, applying this condition, we will show that the upper radical 𝒰(*k) generated by the essential cover *k of the class * of all *-rings has the Amitsur property and 𝒰(*k)(A[x])=𝒰(*k)(A)[x], where a semiprime ring R is called a *-ring if the factor ring R/I is prime radical for every nonzero ideal I of R. The importance of *-rings stems from the fact that a *-ring A is Jacobson semisimple if and only if A is a primitive ring.

scholarly journals Sub-prime radical classes determined by zerorings

1975 ◽  
Vol 12 (1) ◽  
pp. 95-97 ◽  
Author(s):  
B.J. Gardner
Keyword(s):  
Abelian Groups ◽  
Complete Classification ◽  
Prime Radical ◽  
Radical Class

It is shown that the correspondence which associates with each radical class τ of abelian groups the (radical) class of prime radical rings with additive groups in τ gives a complete classification of those radical classes of rings which are determined (as lower radicals) by zerorings.

scholarly journals A uniformly strongly prime radical

1987 ◽  
Vol 43 (1) ◽  
pp. 95-102 ◽  
Author(s):  
D. M. Olson ◽  
R. Lidl
Keyword(s):  
Special Class ◽  
Prime Radical ◽  
Radical Class ◽  
Prime Rings ◽  
The Right

AbstractThe class of all uniformly strongly prime rings is shown to be a special class of rings which generates a radical class which properly contains both the right and left strongly prime radicals and which is independent of the Jacobson and Brown-McCoy radicals.

scholarly journals Radicals and subdirect decompositions of Ω-groups

1990 ◽  
Vol 48 (2) ◽  
pp. 171-198 ◽  
Author(s):  
R. Mlitz ◽  
S. Veldsman
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Radical Class ◽  
Semisimple Class ◽  
Hereditary Radical ◽  
Necessary And Sufficient

AbstractStarting with a class ℳ of Ω-groups, necessary and sufficient conditions on ℳ are given to ensure that the corresponding Hoehnke radical ρ (determined by the subdirect closure of ℳ as semisimple class) is a radical in the sense of Kurosh and Amitsur; has a hereditary semisimple class; satisfies the ADS-property; has a hereditary radical class or satisfies ρN ∩ I ⊆ ρI and lastly, have both a hereditary radical and semisimple class or satisfies ρN ∩ I = ρI.

scholarly journals Prime Ideals and Strongly Prime Ideals of Skew Laurent Polynomial Rings

2008 ◽  
Vol 2008 ◽  
pp. 1-8
Author(s):  
E. Hashemi
Keyword(s):  
Laurent Polynomial ◽  
Prime Radical ◽  
Polynomial Rings ◽  
Prime Ideals ◽  
Laurent Polynomials ◽  
Relationship Of ◽  
The Relationship ◽  
Nil Radical

We first study connections betweenα-compatible ideals ofRand related ideals of the skew Laurent polynomials ringR[x,x−1;α], whereαis an automorphism ofR. Also we investigate the relationship ofP(R)andNr(R)ofRwith the prime radical and the upper nil radical of the skew Laurent polynomial rings. Then by using Jordan's ring, we extend above results to the case whereαis not surjective.

scholarly journals A version of Zhong's coercivity result for a general class of nonsmooth functionals

2002 ◽  
Vol 7 (11) ◽  
pp. 601-612 ◽  
Author(s):  
D. Motreanu ◽  
V. V. Motreanu ◽  
D. Paşca
Keyword(s):  
Asymptotic Behavior ◽  
Variational Principle ◽  
General Result ◽  
General Class ◽  
Lower Semicontinuous ◽  
Ekeland’S Variational Principle ◽  
Ekeland's Variational Principle ◽  
Palais Smale Condition ◽  
Locally Lipschitz ◽  
The Relationship

A version of Zhong's coercivity result (1997) is established for nonsmooth functionals expressed as a sumΦ+Ψ, whereΦis locally Lipschitz andΨis convex, lower semicontinuous, and proper. This is obtained as a consequence of a general result describing the asymptotic behavior of the functions verifying the above structure hypothesis. Our approach relies on a version of Ekeland's variational principle. In proving our coercivity result we make use of a new general Palais-Smale condition. The relationship with other results is discussed.

scholarly journals Phase Equivalent Potentials in Reaction Matrix Theory

1978 ◽  
Vol 31 (2) ◽  
pp. 215
Author(s):  
DW Lang ◽  
JL Cook
Keyword(s):  
General Class ◽  
Matrix Theory ◽  
Matrix Methods ◽  
Reaction Matrix ◽  
Standard Reaction ◽  
The Relationship ◽  
Nonlocal Potentials

The general class of phase equivalent nonlocal potentials is examined using standard reaction matrix methods. The relationship between phase equivalent overlap matrices is derived and it is found that the matrices fall into four general classes.

scholarly journals Statistical convergence of double sequences in fuzzy normed spaces

Filomat ◽  
2012 ◽  
Vol 26 (4) ◽  
pp. 673-681 ◽  
Author(s):  
S.A. Mohiuddine ◽  
H. Şevli ◽  
M. Cancan
Keyword(s):  
Limit Point ◽  
General Class ◽  
Statistical Convergence ◽  
Cluster Point ◽  
Double Sequences ◽  
Normed Spaces ◽  
Statistical Cluster ◽  
Statistical Limit ◽  
Statistical Cluster Point ◽  
The Relationship

In this paper, we study the concepts of statistically convergent and statistically Cauchy double sequences in the framework of fuzzy normed spaces which provide better tool to study a more general class of sequences. We also introduce here statistical limit point and statistical cluster point for double sequences in this framework and discuss the relationship between them.

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