On special radicals, supernilpotent radicals and weakly homomorphically closed classes

1981 ◽  
Vol 31 (2) ◽  
pp. 152-162 ◽  
Author(s):  
Ju. M. Rjabuhin ◽  
R. Wiegandt
Keyword(s):  
Prime Rings ◽  
Special Radical ◽  
Semisimple Class ◽  
Regular Class ◽  
Closed Class ◽  
Proper Subclass ◽  
Semiprime Rings

AbstractIt is proved that a regular essentially closed and weakly homomorphically closed proper subclass of rings consists of semiprime rings. A regular class M defines a supernilpotent upper radical if and only if M consists of semiprime rings and the essential cover Mk of M is contained in the semisimple class S U M. A regular essentially closed class M containing all semisimple prime rings, defines a special upper radical if and only if M satisfies condition (S): every M-ring is a subdirect sum of prime M-rings. Thus we obtained a characterization of semisimple classes of special radicals; a subclas S of rings is the semisimple class of a special radical if and only if S is regular, subdirectly closed, essentially closed, and satisfies condition (S). The results are valid for alternative rings too.

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.

Derivations of differentially semiprime rings

2019 ◽  
Vol 12 (05) ◽  
pp. 1950079
Author(s):  
Ahmad Al Khalaf ◽  
Iman Taha ◽  
Orest D. Artemovych ◽  
Abdullah Aljouiiee
Keyword(s):  
Commutative Ring ◽  
Semiprime Ring ◽  
Prime Rings ◽  
Commutative Case ◽  
Lie Ring ◽  
Lie Rings ◽  
Semiprime Rings ◽  
Torsion Free

Earlier D. A. Jordan, C. R. Jordan and D. S. Passman have investigated the properties of Lie rings Der [Formula: see text] of derivations in a commutative differentially prime rings [Formula: see text]. We study Lie rings Der [Formula: see text] in the non-commutative case and prove that if [Formula: see text] is a [Formula: see text]-torsion-free [Formula: see text]-semiprime ring, then [Formula: see text] is a semiprime Lie ring or [Formula: see text] is a commutative ring.

Semi-Prime Modules

1966 ◽  
Vol 18 ◽  
pp. 823-831 ◽  
Author(s):  
E. H. Feller ◽  
E. W. Swokowski
Keyword(s):  
Prime Rings ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings

Properties and characterizations for prime and semiprime rings have been provided by A. W. Goldie (2, 3). In a previous paper (1), the authors used the results of (2) to characterize prime and uniform prime modules. It is the aim of the present paper to generalize Goldie's work on semi-prime rings (3) to modules. In this setting certain new properties will appear.Notationally, in the work to follow, the symbol R always denotes a ring and all R-modules will be right R-modules.In the theory of rings an ideal C is said to be prime if and only if whenever AB ⊆ C for ideals A and B, then either A ⊆ C or B ⊆ C. A ring is prime if the zero ideal is prime.

Orders in semiprime rings with minimal condition for principal right ideals

1992 ◽  
Vol 121 (3-4) ◽  
pp. 375-380 ◽  
Author(s):  
John Fountain ◽  
Victoria Gould
Keyword(s):  
Minimal Condition ◽  
Alternative Proof ◽  
Prime Rings ◽  
Artinian Rings ◽  
Semiprime Rings

SynopsisAn alternative proof of a theorem which characterises orders in semiprime rings with minimal condition is given. The approach used is to make use of the corresponding result for prime rings and is inspired by Herstein's proof of Goldie's theorem on orders in semisimple Artinian rings.

Centralizing Mappings of Semiprime Rings

1987 ◽  
Vol 30 (1) ◽  
pp. 92-101 ◽  
Author(s):  
H. E. Bell ◽  
W. S. Martindale
Keyword(s):  
Nonempty Subset ◽  
Semiprime Ring ◽  
Prime Rings ◽  
Semiprime Rings

AbstractLet R be a ring with center Z, and S a nonempty subset of R. A mapping F from R to R is called centralizing on S if [x, F(x)] ∊ Z for all x ∊ S. We show that a semiprime ring R must have a nontrivial central ideal if it admits an appropriate endomorphism or derivation which is centralizing on some nontrivial one-sided ideal. Under similar hypotheses, we prove commutativity in prime rings.

scholarly journals Symmetric generalized biderivations on prime rings

2021 ◽  
Vol 39 (4) ◽  
pp. 65-72
Author(s):  
Faiza Shujat
Keyword(s):  
Prime Ring ◽  
Prime Rings ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings

The purpose of the present paper is to prove some results concerning symmetric generalized biderivations on prime and semiprime rings which partially extend some results of Vukman \cite {V}. Infact we prove that: let $R$ be a prime ring of characteristic not two and $I$ be a nonzro ideal of $R$. If $\Delta$ is a symmetric generalized biderivation on $R$ with associated biderivation $D$ such that $[\Delta(x,x), \Delta(y,y)]=0$ for all $x,y \in I$, then one of the following conditions hold\\ \begin{enumerate} \item $R$ is commutative. \item $\Delta$ acts as a left bimultiplier on $R$. \end{enumerate}

scholarly journals A characterization of $ b- $generalized derivations on prime rings with involution

2021 ◽  
Vol 7 (2) ◽  
pp. 2413-2426
Author(s):  
Mohd Arif Raza ◽  
◽  
Abdul Nadim Khan ◽  
Husain Alhazmi ◽  
Keyword(s):  
Generalized Derivations ◽  
Prime Rings ◽  
Functional Identities ◽  
Rings With Involution

<abstract><p>In this note, we characterize $ b- $generalized derivations which are strong commutative preserving (SCP) on $ \mathscr{R} $. Moreover, we also discuss and characterize $ b- $generalized derivations involving certain $ \ast- $differential/functional identities on rings possessing involution.</p></abstract>

Commutative Endomorphism Rings

1971 ◽  
Vol 23 (1) ◽  
pp. 69-76 ◽  
Author(s):  
J. Zelmanowitz
Keyword(s):  
Abelian Group ◽  
Cyclic Group ◽  
Endomorphism Ring ◽  
Endomorphism Rings ◽  
Prime Rings ◽  
Infinite Cyclic Group ◽  
Free Abelian Groups ◽  
Scalar Action ◽  
Torsion Free

The problem of classifying the torsion-free abelian groups with commutative endomorphism rings appears as Fuchs’ problems in [4, Problems 46 and 47]. They are far from solved, and the obstacles to a solution appear formidable (see [4; 5]). It is, however, easy to see that the only dualizable abelian group with a commutative endomorphism ring is the infinite cyclic group. (An R-module Miscalled dualizable if HomR(M, R) ≠ 0.) Motivated by this, we study the class of prime rings R which possess a dualizable module M with a commutative endomorphism ring. A characterization of such rings is obtained in § 6, which as would be expected, places stringent restrictions on the ring and the module.Throughout we will write homomorphisms of modules on the side opposite to the scalar action. Rings will not be assumed to contain identity elements unless otherwise indicated.

On Prime and Semiprime Rings with Derivations

2006 ◽  
Vol 13 (03) ◽  
pp. 371-380 ◽  
Author(s):  
Nurcan Argaç
Keyword(s):  
Prime Ring ◽  
Nonempty Subset ◽  
Semiprime Ring ◽  
Nonzero Ideal ◽  
Prime Rings ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Torsion Free

Let R be a ring and S a nonempty subset of R. A mapping f: R → R is called commuting on S if [f(x),x] = 0 for all x ∈ S. In this paper, firstly, we generalize the well-known result of Posner related to commuting derivations on prime rings. Secondly, we show that if R is a semiprime ring and I is a nonzero ideal of R, then a derivation d of R is commuting on I if one of the following conditions holds: (i) For all x, y ∈ I, either d([x,y]) = [x,y] or d([x,y]) = -[x,y]. (ii) For all x, y ∈ I, either d(x ◦ y) = x ◦ y or d(x ◦ y) = -(x ◦ y). (iii) R is 2-torsion free, and for all x, y ∈ I, either [d(x),d(y)] = d([x,y]) or [d(x),d(y)] = d([y,x]). Furthermore, if d(I) ≠ {0}, then R has a nonzero central ideal. Finally, we introduce the notation of generalized biderivation and prove that every generalized biderivation on a noncommutative prime ring is a biderivation.

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