scholarly journals Strongly prime near-rings

1988 ◽  
Vol 31 (3) ◽  
pp. 337-343 ◽  
Author(s):  
N. J. Groenewald
Keyword(s):  
Special Class ◽  
Prime Ideals ◽  
Prime Rings

Strongly prime rings were introduced by Handelman and Lawrence [5] and in [2] Groenewald and Heyman investigated the upper radical determined by the class of all strongly prime rings. In this paper we extend the concept of strongly prime to near-rings. We show that the class M of distributively generated near-rings is a special class in the sense of Kaarli [6]. We also show that if N is any distributively generated near-ring, then UM(N), UM denotes the upper radical determined by the class M, coincides with the intersection of all the strongly prime ideals of N.

scholarly journals On generalised prime essential rings and special and nonspecial radicals

2007 ◽  
Vol 76 (2) ◽  
pp. 263-268 ◽  
Author(s):  
Halina France-Jackson
Keyword(s):  
Special Class ◽  
Prime Rings ◽  
Semisimple Class ◽  
Semiprime Rings

For a supernilpotent radical α and a special class σ of rings we call a ring R (α, σ)-essential if R is α-semisimple and for each ideal P of R with R/P ε σ, P ∩ I ≠ 0 whenever I is a nonzero two-sided ideal of R. (α, σ)-essential rings form a generalisation of prime essential rings introduced by L. H. Rowen in his study of semiprime rings and their subdirect decompositions and they have been a subject of investigations of many prominent authors since. We show that many important results concerning prime essential rings are also valid for (α, σ)-essential rings and demonstrate how (α, σ)-essential rings can be used to determine whether a supernilpotent radical is special. We construct infinitely many supernilpotent nonspecial radicals whose semisimple class of prime rings is zero and show that such radicals form a sublattice of the lattice of all supernilpotent radicals. This generalises Yu.M. Ryabukhin's example.

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 Prime rings for which the set of nonzero ideals is a special class

1991 ◽  
Vol 51 (1) ◽  
pp. 27-32 ◽  
Author(s):  
B. J. Gardner
Keyword(s):  
Principal Ideal ◽  
Special Class ◽  
Subdirectly Irreducible ◽  
Prime Rings ◽  
Ring Homomorphisms ◽  
Principal Ideal Domains ◽  
Associative Rings ◽  
The Continuum

AbstractWe show that the rings described in the title are precisely the indecomposable injectives for the category whose objects are the associative rings and whose morphisms are the ring homomorphisms with accessible images. These rings are more or less completely known. Those of cardinality greater than that of the continuum are subdirectly irreducible but there are some nontrivial principal ideal domains in the class.

scholarly journals A simple ring separating certain radicals

1975 ◽  
Vol 16 (1) ◽  
pp. 29-31 ◽  
Author(s):  
G. A. P. Heyman ◽  
W. G. Leavitt
Keyword(s):  
Homomorphic Image ◽  
Special Class ◽  
Radical Class ◽  
Subdirectly Irreducible ◽  
Prime Rings ◽  
Simple Ring ◽  
Special Radical ◽  
Hereditary Class

All rings considered will be associative. For a class M of rings let UM be the class of all rings having no non-zero homomorphic image in M. A hereditary class M of prime rings is called a “special class” [see 1, p. 191] if it has the property that when I ∈ M with I an ideal of a ring R, then R/I* ∈ Mwhere I* is the annihilator of I in R, and the corresponding radical class UM is then a “special radical”. Let S be the class of all subdirectly irreducible rings with simple heart.

scholarly journals Essential extensions in radical theory for rings

1977 ◽  
Vol 23 (3) ◽  
pp. 340-347 ◽  
Author(s):  
G. A. P. Heyman ◽  
C. Roos
Keyword(s):  
Special Class ◽  
Intersection Property ◽  
Prime Rings ◽  
Radical Theory ◽  
Hereditary Class ◽  
Closed Class ◽  
Essential Extensions

AbstractThe essential cover Mk of a class M is defined as the class of all essential extensions of rings belonging to M. M is called essentially closed if Mk = M. Every class M has a unique essential closure, i.e. a smallest essentially closed class containing M.Let M be a hereditary class of (semi)prime rings. Then M is proved to be a (weakly) special class if and only if M is essentially closed. A main result is that Mk is the smallest (weakly) special class containing M. Further it is shown that the upper radical UM determined by M, is hereditary if and only if UM has the intersection property with respect to Mk.

scholarly journals Prime and special ideals in structural matrix rings over a ring without unity

1988 ◽  
Vol 45 (2) ◽  
pp. 220-226 ◽  
Author(s):  
L. Van Wyk
Keyword(s):  
Special Class ◽  
Associative Ring ◽  
Matrix Ring ◽  
Boolean Matrix ◽  
Prime Ideals ◽  
Matrix Rings ◽  
Quasi Order ◽  
Structural Matrix Ring ◽  
Complete Matrix ◽  
Structural Matrix

AbstractA. D. Sands showed that there is a 1–1 correspondence between the prime ideals of an arbitraty associative ring R and the complete matrix ring Mn(R) via P→ Mn(P). A structural matrix ring M(B, R) is the ring of all n × n matrices over R with 0 in the positions where the n × n boolean matrix B, B a quasi-order, has 0. The author characterized the special ideals of M(B, R′), in case R′ has unity, for certain special lasses of rings. In this note results of sands and the author are generalized to structural matrix rings over rings without unity. I t turns out that, although the class of prime simple rings is not a special class, Nagata's M-radical has the same form in structural matrix rings as the special radicals studied by the author.

On the Periodic Radical of a Ring

1995 ◽  
Vol 38 (2) ◽  
pp. 215-217 ◽  
Author(s):  
Xiuzhan Guo
Keyword(s):  
Subdirect Product ◽  
Prime Ideals ◽  
Prime Rings

AbstractLet R be a ring and P(R) the sum of all periodic ideals of R. We prove that P(R) is the intersection of all prime ideals Pα such that contains no nontrivial periodic ideals. We also prove that P(R) = 0 if and only if Rs is a subdirect product of prime rings Rα with P(Rα) = 0.

scholarly journals On special atoms

1998 ◽  
Vol 64 (3) ◽  
pp. 302-306 ◽  
Author(s):  
H. France Jackson
Keyword(s):  
Prime Ring ◽  
Special Class ◽  
Prime Radical ◽  
Prime Rings

AbstractA characterization of all special atoms in the from of the upper radical generated by the class of all prime rings outside the smallest special class containing some prime ring is provided and prime rings for which the above mentioned upper radical coincides with the prime radical are investigated.

scholarly journals Normal radicals and normal classes of modules

1988 ◽  
Vol 30 (1) ◽  
pp. 97-100
Author(s):  
W. K. Nicholson ◽  
J. F. Watters
Keyword(s):  
Special Class ◽  
Prime Rings ◽  
Special Classes

The study of special radicals was begun by Andrunakievič [1]. A class of prime rings is called special if it is hereditary and closed under prime extensions. The upper radicals determined by special classes are called special. In later works Andrunakievič and Rjabuhin [2] and [3] defined the concept of a special class of modules.

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