• 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.

• 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.

• 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.

• Journal of the Korean Mathematical Society
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• v.50 no.5
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• pp.1051-1066
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• 2013

By utilizing known characterizations of ${\omega}$-Noetherian rings in terms of injective modules, we give more characterizations of ${\omega}$-Noetherian rings. More precisely, we show that a commutative ring R is ${\omega}$-Noetherian if and only if the direct limit of GV -torsion-free injective R-modules is injective; if and only if every R-module has a GV -torsion-free injective (pre)cover; if and only if the direct sum of injective envelopes of ${\omega}$-simple R-modules is injective; if and only if the essential extension of the direct sum of GV -torsion-free injective R-modules is the direct sum of GV -torsion-free injective R-modules; if and only if every $\mathfrak{F}_{w,f}(R)$-injective ${\omega}$-module is injective; if and only if every GV-torsion-free R-module admits an $i$-decomposition.

• Bulletin of the Korean Mathematical Society
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• v.39 no.3
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• pp.485-494
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• 2002

In this paper we will show that if there exist derivations D, G on a n!-torsion free semi-prime ring R such that the mapping $D^2+G$ is n-commuting on R, then D and G are both commuting on R. And we shall give the algebraic conditions on a ring that a Jordan derivation is zero.

• Honam Mathematical Journal
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• v.28 no.4
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• pp.499-511
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• 2006

It is well-known that every finitely generated torsion-free module over a principal ideal domain is free. This will be generalized. We deal with ideals of the finite, external direct product of certain rings. Finally, if M is a torsion-free, finitely generated module over a reduced, Noetherian ring A, then we prove that Ms is a projective module over As, where $S=A{\setminus}(A)$.

• Bulletin of the Korean Mathematical Society
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• v.59 no.5
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• pp.1289-1304
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• 2022

In this note, we show that a strongly 𝜙-ring R is a 𝜙-PvMR if and only if any 𝜙-torsion-free R-module is 𝜙-w-flat, if and only if any GV-torsion-free divisible R-module is nonnil-absolutely w-pure, if and only if any GV-torsion-free h-divisible R-module is nonnil-absolutely w-pure, if and only if any finitely generated nonnil ideal of R is w-projective.

• Bulletin of the Korean Mathematical Society
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• v.46 no.3
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• pp.553-565
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• 2009

In this paper we prove that every generalized Jordan triple higher derivation on a 2-torsion free semiprime ring is a generalized higher derivation. This extend the main result of [9] to the case of a semiprime ring.

• Bulletin of the Korean Mathematical Society
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• v.46 no.5
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• pp.857-866
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• 2009

Let n be a fixed positive integer, R be a 2n!-torsion free prime ring and $\mu$, $\nu$ be a pair of generalized derivations on R. If < $\mu^2(x)+\nu(x),\;x^n$ > = 0 for all x $\in$ R, then $\mu$ and $\nu$ are either left multipliers or right multipliers. Let n be a fixed positive integer, R be a noncommutative 2n!-torsion free prime ring with the center $C_R$ and d, g be a pair of derivations on R. If < $d^2(x)+g(x)$, $x^n$ > $\in$ $C_R$ for all x $\in$ R, then d = g = 0. Then we apply these purely algebraic techniques to obtain several range inclusion results of pair of (generalized-)derivations on a Banach algebra.

• Bulletin of the Korean Mathematical Society
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• v.39 no.4
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• pp.635-643
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• 2002

Our main goal is to show that if there exist Jordan derivations D and G on a noncommutative (n + 1)!-torsion free prime ring R such that $$D(x)x^n-x^nG(x)\in\ C(R)$$ for all $x\in\ R$, then we have D=0 and G=0. We also prove that if there exists a derivation D on a noncommutative 2-torsion free prime ring R such that the mapping $\chi$longrightarrow[aD($\chi$), $\chi$] is commuting on R, then we have either a = 0 or D = 0.