• Title, Summary, Keyword: prime and semiprime rings

### FULLY PRIME MODULES AND FULLY SEMIPRIME MODULES

• Beachy, John A.;Medina-Barcenas, Mauricio
• Bulletin of the Korean Mathematical Society
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• v.57 no.5
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• pp.1177-1193
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• 2020
• Fully prime rings (in which every proper ideal is prime) have been studied by Blair and Tsutsui, and fully semiprime rings (in which every proper ideal is semiprime) have been studied by Courter. For a given module M, we introduce the notions of a fully prime module and a fully semiprime module, and extend certain results of Blair, Tsutsui, and Courter to the category subgenerated by M. We also consider the relationship between the conditions (1) M is a fully prime (semiprime) module, and (2) the endomorphism ring of M is a fully prime (semiprime) ring.

### SOME CONDITIONS ON DERIVATIONS IN PRIME NEAR-RINGS

• Cho, Yong-Uk
• The Pure and Applied Mathematics
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• v.8 no.2
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• pp.145-152
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• 2001
• Posner [Proc. Amer. Math. Soc. 8 (1957), 1093-1100] defined a derivation on prime rings and Herstein [Canad, Math. Bull. 21 (1978), 369-370] derived commutative property of prime ring with derivations. Recently, Bergen [Canad. Math. Bull. 26 (1983), 267-227], Bell and Daif [Acta. Math. Hunger. 66 (1995), 337-343] studied derivations in primes and semiprime rings. Also, in near-ring theory, Bell and Mason [Near-Rungs and Near-Fields (pp. 31-35), Proceedings of the conference held at the University of Tubingen, 1985. Noth-Holland, Amsterdam, 1987; Math. J. Okayama Univ. 34 (1992), 135-144] and Cho [Pusan Kyongnam Math. J. 12 (1996), no. 1, 63-69] researched derivations in prime and semiprime near-rings. In this paper, Posner, Bell and Mason＇s results are extended in prime near-rings with some conditions.

### SEMIPRIME NEAR-RINGS WITH ORTHOGONAL DERIVATIONS

• Park, Kyoo-Hong;Jung, Yong-Soo
• The Pure and Applied Mathematics
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• v.13 no.4
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• pp.303-310
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• 2006
• M. $Bre\v{s}ar$ and J. Vukman obtained some results concerning orthogonal derivations in semiprime rings which are related to the result that is well-known to a theorem of Posner for the product of two derivations in prime rings. In this paper, we present orthogonal generalized derivations in semiprime near-rings.

### GENERALIZED DERIVATIONS AND DERIVATIONS OF RINGS AND BANACH ALGEBRAS

• Jung, Yong-Soo
• Honam Mathematical Journal
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• v.35 no.4
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• pp.625-637
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• 2013
• We investigate anti-centralizing and skew-centralizing mappings involving generalized derivations and derivations on prime and semiprime rings. We also obtain some range inclusion results for generalized linear derivations and linear derivations on Banach algebras by applying the algebraic techniques. Some results in this note are to improve the ones in [22].

### STRONG COMMUTATIVITY PRESERVING MAPPINGS ON SEMIPRIME RINGS

• Bulletin of the Korean Mathematical Society
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• v.43 no.4
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• pp.711-713
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• 2006
• Let R be a semiprime ring and f be an endomorphism on R. If f is a strong commutativity preserving (simply, scp) map on a non-zero ideal U of R, then f is commuting on U.

### DERIVATIONS OF PRIME AND SEMIPRIME RINGS

• Argac, Nurcan;Inceboz, Hulya G.
• Journal of the Korean Mathematical Society
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• v.46 no.5
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• pp.997-1005
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• 2009
• Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and n a fixed positive integer. (i) If (d(x)y+xd(y)+d(y)x+$yd(x))^n$ = xy + yx for all x, y $\in$ I, then R is commutative. (ii) If char R $\neq$ = 2 and (d(x)y + xd(y) + d(y)x + $yd(x))^n$ - (xy + yx) is central for all x, y $\in$ I, then R is commutative. We also examine the case where R is a semiprime ring.

### On Orthogonal Generalized (σ, τ)-Derivations of Semiprim Near-Rings

• Huang, Shuliang
• Kyungpook Mathematical Journal
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• v.50 no.3
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• pp.379-387
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• 2010
• In this paper, we present some results concerning orthogonal generalized (${\sigma},{\tau}$)-derivations in semiprime near-rings. These results are a generalization of result of Bresar and Vukman, which are related to a theorem of Posner for the product of two derivations in prime rings.

### ON 4-PERMUTING 4-DERIVATIONS IN PRIME AND SEMIPRIME RINGS

• Park, Kyoo-Hong
• The Pure and Applied Mathematics
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• v.14 no.4
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• pp.271-278
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• 2007
• Let R be a 2-torsion free semiprime ring. Suppose that there exists a 4-permuting 4-derivation ${\Delta}:R{\times}R{\times}R{\times}R{\rightarrow}R$ such that the trace is centralizing on R. Then the trace of ${\Delta}$ is commuting on R. In particular, if R is a 3!-torsion free prime ring and ${\Delta}$ is nonzero under the same condition, then R is commutative.

### ON PRIME AND SEMIPRIME RINGS WITH SYMMETRIC n-DERIVATIONS

• Park, Kyoo-Hong
• Journal of the Chungcheong Mathematical Society
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• v.22 no.3
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• pp.451-458
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• 2009
• Let $n{\geq}2$ be a fixed positive integer and let R be a noncommutative n!-torsion free semiprime ring. Suppose that there exists a symmetric n-derivation $\Delta$ : $R^{n}{\rightarrow}R$ such that the trace of $\Delta$ is centralizing on R. Then the trace is commuting on R. If R is a n!-torsion free prime ring and $\Delta{\neq}0$ under the same condition. Then R is commutative.

### ON PRIME AND SEMIPRIME RINGS WITH PERMUTING 3-DERIVATIONS

• Jung, Yong-Soo;Park, Kyoo-Hong
• Bulletin of the Korean Mathematical Society
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• v.44 no.4
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• pp.789-794
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• 2007
• Let R be a 3-torsion free semiprime ring and let I be a nonzero two-sided ideal of R. Suppose that there exists a permuting 3-derivation ${\Delta}:R{\times}R{\times}R{\rightarrow}R$ such that the trace is centralizing on I. Then the trace of ${\Delta}$ is commuting on I. In particular, if R is a 3!-torsion free prime ring and ${\Delta}$ is nonzero under the same condition, then R is commutative.