• Kyungpook Mathematical Journal
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• v.51 no.3
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• pp.301-309
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• 2011

In this paper we dene the notions of left *-bimultiplier, *-bimultiplier and generalized *-biderivation, and to prove that if a semiprime *-ring admits a left *-bimultiplier M, then M maps R ${\times}$ R into Z(R). In Section 3, we discuss the applications of theory of *-bimultipliers. Further, it was shown that if a semiprime *-ring R admits a symmetric generalized *-biderivation G : R ${\times}$ R ${\rightarrow}$ R with an associated nonzero symmetric *-biderivation R ${\times}$ R ${\rightarrow}$ R, then G maps R ${\times}$ R into Z(R). As an application, we establish corresponding results in the setting of $C^*$-algebra.

• Journal of the Chungcheong Mathematical Society
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• v.33 no.3
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• pp.307-318
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• 2020

In this paper, we extend the notion of a generalized derivation F associated with derivation d to two generalized derivations F and G associated with the same derivation d, as a new idea, to obtain the commutativity of prime rings under certain conditions.

• Communications of the Korean Mathematical Society
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• v.27 no.1
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• pp.69-76
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• 2012

Let R be a ring, and ${\alpha}$ be an endomorphism of R. An additive mapping H : R ${\rightarrow}$ R is called a left ${\alpha}$-multiplier (centralizer) if H(xy) = H(x)${\alpha}$(y) holds for all x,y $\in$ R. In this paper, we shall investigate the commutativity of prime and semiprime rings admitting left ${\alpha}$-multiplier satisfying any one of the properties: (i) H([x,y])-[x,y] = 0, (ii) H([x,y])+[x,y] = 0, (iii) $H(x{\circ}y)-x{\circ}y=0$, (iv) $H(x{\circ}y)+x{\circ}y=0$, (v) H(xy) = xy, (vi) H(xy) = yx, (vii) $H(x^2)=x^2$, (viii) $H(x^2)=-x^2$ for all x, y in some appropriate subset of R.

• 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.

• Journal of the Chungcheong Mathematical Society
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• v.29 no.4
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• pp.531-542
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• 2016

Let R be a 3!-torsion free noncommutative semiprime ring, and suppose there exists a Jordan derivation $D:R{\rightarrow}R$ such that [[D(x),x], x]D(x) = 0 or D(x)[[D(x), x], x] = 0 for all $x{\in}R$. In this case we have $[D(x),x]^3=0$ for all $x{\in}R$. Let A be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation $D:A{\rightarrow}A$ such that $[[D(x),x],x]D(x){\in}rad(A)$ or $D(x)[[D(x),x],x]{\in}rad(A)$ for all $x{\in}A$. In this case, we show that $D(A){\subseteq}rad(A)$.

• Journal of the Chungcheong Mathematical Society
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• v.31 no.1
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• pp.1-12
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• 2018

Let R be a 3!-torsion free noncommutative semiprime ring, and suppose there exists a Jordan derivation $D:R{\rightarrow}R$ such that $D^2(x)[D(x),x]=0$ or $[D(x),x]D^2(x)=0$ for all $x{\in}R$. In this case we have $f(x)^5=0$ for all $x{\in}R$. Let A be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation $D:A{\rightarrow}A$ such that $D^2(x)[D(x),x]{\in}rad(A)$ or $[D(x),x]D^2(x){\in}rad(A)$ for all $x{\in}A$. In this case, we show that $D(A){\subseteq}rad(A)$.

• Bulletin of the Korean Mathematical Society
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• v.38 no.4
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• pp.623-633
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• 2001

In this note we study the prime radicals of formal power series rings, and the shapes of them under the condition that the prime radical is nilpotent. Furthermore we observe the condition structurally, adding related examples to the situations that occur naturally in the process.

• Journal of the Korean Mathematical Society
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• v.50 no.6
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• pp.1271-1290
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• 2013

Let M be a fixed left R-module. For a left R-module X, we introduce the notion of M-prime (resp. M-semiprime) submodule of X such that in the case M=R, it coincides with prime (resp. semiprime) submodule of X. Other concepts encountered in the general theory are M-$m$-system sets, M-$n$-system sets, M-prime radical and M-Baer's lower nilradical of modules. Relationships between these concepts and basic properties are established. In particular, we identify certain submodules of M, called "primeM-ideals", that play a role analogous to that of prime (two-sided) ideals in the ring R. Using this definition, we show that if M satisfies condition H (defined later) and $Hom_R(M,X){\neq}0$ for all modules X in the category ${\sigma}[M]$, then there is a one-to-one correspondence between isomorphism classes of indecomposable M-injective modules in ${\sigma}[M]$ and prime M-ideals of M. Also, we investigate the prime M-ideals, M-prime submodules and M-prime radical of Artinian modules.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.51 no.2
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• pp.579-594
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• 2014

Let R be a ring and $Nil_*$(R) be the prime radical of R. In this paper, we say that a ring R is left $Nil_*$-coherent if $Nil_*$(R) is coherent as a left R-module. The concept is introduced as the generalization of left J-coherent rings and semiprime rings. Some properties of $Nil_*$-coherent rings are also studied in terms of N-injective modules and N-flat modules.