Quasi-critical ring of a primitive ring with nonzero socle

1997 ◽  
Vol 42 (3) ◽  
pp. 189-192
Author(s):  
Zimei Jiang
Keyword(s):  
Critical Ring ◽  
Primitive Ring

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 Rings with a finite set of nonnilpotents

1979 ◽  
Vol 2 (1) ◽  
pp. 121-126 ◽  
Author(s):  
Mohan S. Putcha ◽  
Adil Yaqub
Keyword(s):  
Division Ring ◽  
Nonnegative Integer ◽  
Finite Ring ◽  
Structure Theory ◽  
Nilpotent Elements ◽  
Finite Set ◽  
Semisimple Ring ◽  
Nil Ring ◽  
Primitive Ring

LetRbe a ring and letNdenote the set of nilpotent elements ofR. Letnbe a nonnegative integer. The ringRis called aθn-ring if the number of elements inRwhich are not inNis at mostn. The following theorem is proved: IfRis aθn-ring, thenRis nil orRis finite. Conversely, ifRis a nil ring or a finite ring, thenRis aθn-ring for somen. The proof of this theorem uses the structure theory of rings, beginning with the division ring case, followed by the primitive ring case, and then the semisimple ring case. Finally, the general case is considered.

scholarly journals A Note on a Self Injective Ring

1965 ◽  
Vol 8 (1) ◽  
pp. 29-32 ◽  
Author(s):  
Kwangil Koh
Keyword(s):  
Left Ideal ◽  
Prime Ring ◽  
Minimum Condition ◽  
Maximum Condition ◽  
Simple Ring ◽  
The Right ◽  
Primitive Ring

A ring R with unity is called right (left) self injective if the right (left) R-module R is injective [7]. The purpose of this note is to prove the following: Let R be a prime ring with a maximal annihilator right (left) ideal. If R is right (left) self injective then R is a primitive ring with a minimal one-sided ideal. If R satisfies the maximum condition on annihilator right (left) ideals and R is right (left) self injective then R is a simple ring with the minimum condition on one-sided ideals.

scholarly journals A commutativity theorem for semi-primitive rings

1986 ◽  
Vol 34 (2) ◽  
pp. 293-295 ◽  
Author(s):  
Hisao Tominaga
Keyword(s):  
Commutativity Theorem ◽  
Positive Integers ◽  
Primitive Ring

In this brief note, we prove the following: Let R be a semi-primitive ring. Suppose that for each pair x, y ε R there exist positive integers m = m (x,y) and n = n (x,y) such that either [xm,(xy) n − (yx) n] = 0 or [xm,(xy) n + (yx) n] = 0. Then R is commutative.

scholarly journals The Mappilla Rebellion, 1921: Peasant Revolt in Malabar

1977 ◽  
Vol 11 (1) ◽  
pp. 57-99 ◽  
Author(s):  
Robert L. Hardgrave
Keyword(s):  
Human Society ◽  
Historical Processes ◽  
Peasant Revolt ◽  
Critical Ring ◽  
The Way

In any society the dominant groups are the ones with the most to hide about the way society works. Very often therefore truthful analyses are bound to have a critical ring, to seem like exposures rather than objective statements, … For all students of human society, sympathy with the victims of historical processes and skepticism about the victors' claims provide essential safeguards against being taken in by the dominant mythology. A scholar who tries to be objective needs these feelings as part of his ordinary working equipment.

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