• Journal for History of Mathematics
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• v.12 no.2
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• pp.151-158
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• 1999

In this paper, the concepts of semiprime $L$-fuzzy ideals and semiprimary $L$-fuzzy ideals of a $L$-fuzzy ring are introduced and some fundamental propositions proved. And we investigate the relation between fuzzy nil radical and semiprimary.

• Communications of the Korean Mathematical Society
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• v.17 no.4
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• pp.607-618
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• 2002

Let A be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation D : A \longrightarrowA such that D($\chi$)[D($\chi$),$\chi$]D($\chi$) $\in$ rad(A) for all $\chi$ A. In this case, we have D(A) ⊆ rad(A).

• 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.

• The Pure and Applied Mathematics
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• v.15 no.3
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• pp.259-296
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• 2008

Let A be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation D : A $\rightarrow$ A such that $D(x)^2$[D(x),x] $\in$ rad(A) or [D(x),x]$D(x)^2$ $\in$ rad(A) for all x $\in$ A. In this case, we have D(A) $\subseteq$ rad(A).

• The Pure and Applied Mathematics
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• v.15 no.2
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• pp.179-201
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• 2008

Let A be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation $D\;:\;A{\rightarrow}A$ such that $D(x)[D(x),x]^2\;{\in}\;rad(A)$ or $[D(x), x]^2 D(x)\;{\in}\;rad(A)$ for all $x\;{\in}\ A$. In this case, we have $D(A)\;{\subseteq}\;rad(A)$.

• Communications of the Korean Mathematical Society
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• v.34 no.3
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• pp.811-818
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• 2019

Let A be a Banach algebra with rad(A). We show that if there exists a continuous linear Jordan derivation D on A, then $$[D(x),x]D(x)^2{\in}rad(A)$$ if and only if $D(x)[D(x),x]D(x){\in}rad(A)$ for all $x{\in}A$.

• Journal of applied mathematics & informatics
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• v.7 no.1
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• pp.305-317
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• 2000

It is well-known that every derivation on a commutative Banach algebra maps into its radical. In this paper we shall give the various algebraic conditions on the ring that every Jordan derivation on a noncommutative ring with suitable characteristic conditions is zero and using this result, we show that every continuous linear Jordan derivation on a noncommutative Banach algebra maps into its radical under the suitable conditions.

• Bulletin of the Korean Mathematical Society
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• v.56 no.1
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• pp.23-29
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• 2019

Rings in which all accessible subrings are ideals are called filial. A ring R is called fully filial if all its subrings are filial (that is rings in which the relation of being an ideal is transitive). The present paper is devoted to the study of fully filial torsion rings. We prove a classification theorem for semiprime fully filial torsion rings.

• Communications of the Korean Mathematical Society
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• v.32 no.3
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• pp.535-542
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• 2017

Let R be an associative ring and ${\alpha},{\beta}:R{\rightarrow}R$ ring homomorphisms. An additive mapping $d:R{\rightarrow}R$ is called an (${\alpha},{\beta}$)-derivation of R if $d(xy)=d(x){\alpha}(y)+{\beta}(x)d(y)$ is fulfilled for any $x,y{\in}R$, and an additive mapping $D:R{\rightarrow}R$ is called a generalized (${\alpha},{\beta}$)-derivation of R associated with an (${\alpha},{\beta}$)-derivation d if $D(xy)=D(x){\alpha}(y)+{\beta}(x)d(y)$ is fulfilled for all $x,y{\in}R$. In this note, we intend to generalize a theorem of Vukman [5], and a theorem of Daif and El-Sayiad [2].

• 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.