• 한국수학교육학회지시리즈B:순수및응용수학
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• 제23권4호
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• pp.347-375
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• 2016

Let R be a 3!-torsion free noncommutative semiprime ring, U a Lie ideal of R, and let $D:R{\rightarrow}R$ be a Jordan derivation. If [D(x), x]D(x) = 0 for all $x{\in}U$, then D(x)[D(x), x]y - yD(x)[D(x), x] = 0 for all $x,y{\in}U$. And also, if D(x)[D(x), x] = 0 for all $x{\in}U$, then [D(x), x]D(x)y - y[D(x), x]D(x) = 0 for all $x,y{\in}U$. And we shall give their applications in Banach algebras.

• 충청수학회지
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• 제29권4호
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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 $D(A){\subseteq}rad(A)$.

• 대한수학회보
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• 제50권6호
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• pp.1905-1914
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• 2013

Let D be an integral domain with quotient field K, M a torsion-free D-module, X an indeterminate, and $N_v=\{f{\in}D[X]|c(f)_v=D\}$. Let $q(M)=M{\otimes}_D\;K$ and $M_{w_D}$={$x{\in}q(M)|xJ{\subseteq}M$ for a nonzero finitely generated ideal J of D with $J_v$ = D}. In this paper, we show that $M_{w_D}=M[X]_{N_v}{\cap}q(M)$ and $(M[X])_{w_{D[X]}}{\cap}q(M)[X]=M_{w_D}[X]=M[X]_{N_v}{\cap}q(M)[X]$. Using these results, we prove that M is a strong Mori D-module if and only if M[X] is a strong Mori D[X]-module if and only if $M[X]_{N_v}$ is a Noetherian $D[X]_{N_v}$-module. This is a generalization of the fact that D is a strong Mori domain if and only if D[X] is a strong Mori domain if and only if $D[X]_{N_v}$ is a Noetherian domain.

• 대한수학회지
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• 제48권1호
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• pp.49-61
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• 2011

Let D be an integral domain with quotient field K, X be a nonempty set of indeterminates over D, * be a star operation on D, $N_*$={f $\in$ D[X]|c(f)$^*$= D}, $*_w$ be the star operation on D defined by $I^{*_w}$ = ID[X]${_N}_*$ $\cap$ K, and [*] be the star operation on D[X] canonically associated to * as in Theorem 2.1. Let $A^g$ (resp., $A^{[*]g}$, $A^{[*]g}$) be the global (resp.,*-global, [*]-global) transform of a ring A. We show that D is a $*_w$-Noetherian domain if and only if D[X] is a [*]-Noetherian domain. We prove that $D^{*g}$[X]${_N}_*$ = (D[X]${_N}_*$)$^g$ = (D[X])$^{[*]g}$; hence if D is a $*_w$-Noetherian domain, then each ring between D[X]${_N}_*$ and $D^{*g}$[X]${_N}_*$ is a Noetherian domain. Let $\tilde{D}$ = $\cap${$D_P$|P $\in$ $*_w$-Max(D) and htP $\geq$2}. We show that $D\;\subseteq\;\tilde{D}\;\subseteq\;D^{*g}$ and study some properties of $\tilde{D}$ and $D^{*g}$.

• 충청수학회지
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• 제31권1호
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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 $D(A){\subseteq}rad(A)$.

• 대한수학회논문집
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• 제34권3호
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• pp.951-965
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• 2019

Let $f:X{\rightarrow}X$ be a continuous map on a compact metric space (X, d) and for an arbitrary $x{\in}X$, $${\mathcal{SC}}_d(x,f):=\{y{\mid}x{\text{ can be strong }}d-{\text{chain to }}y\}$$. We give an example to show that ${\mathcal{SC}}_d(x,f)$ is dependent on the metric d on X but it is a closed and f-invariant set. We prove that if ${\mathcal{SC}}_d(x,f){\supseteq}{\Omega}(f)$ or f has the asymptotic-average shadowing property, then ${\mathcal{SC}}_d(x,f)=X$. Also, we show that if f has the shadowing property, then ${\lim}\;{\sup}_{n{\in}{\mathbb{N}}}\{f^n\}={\mathcal{SC}}_d(f)$ where ${\mathcal{SC}}_d(f)=\{(x,y){\mid}y{\in}{\mathcal{SC}}_d(x,f)\}$. For each $n{\in}{\mathbb{N}}$, we give an example in which ${\mathcal{SCR}}_d(f^n){\neq}{\mathcal{SCR}}_d(f)$. In spite of it, we prove that if $f^{-1}:(X,d){\rightarrow}(X,d)$ is an equicontinuous map, then ${\mathcal{SCR}}_d(f^n)={\mathcal{SCR}}_d(f)$ for all $n{\in}{\mathbb{N}}$.

• Korean Journal of Mathematics
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• 제24권3호
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• pp.537-543
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• 2016

Let D be an integral domain, X be an indeterminate over D, D[X] be the polynomial ring over D, and D(X) be the Nagata ring of D. Let [d] be the star operation on D[X], which is an extension of the d-operation on D as in [5, Theorem 2.3]. In this paper, we show that D is a sharp domain if and only if D[X] is a [d]-sharp domain, if and only if D(X) is a sharp domain.

• Kyungpook Mathematical Journal
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• 제55권1호
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• pp.1-12
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• 2015

Let R be a ring and $D:R^n{\longrightarrow}R$ be n-additive mapping. A map $d:R{\longrightarrow}R$ is said to be the trace of D if $d(x)=D(x,x,{\ldots}x)$ for all $x{\in}R$. Suppose that ${\alpha},{\beta}$ are endomorphisms of R. For any $a,b{\in}R$, let < a, b > $_{({\alpha},{\beta})}=a{\alpha}(b)+{\beta}(b)a$. In the present paper under certain suitable torsion restrictions it is shown that D = 0 if R satisfies either < d(x), $x^m$ > $_{({\alpha},{\beta})}=0$, for all $x{\in}R$ or ${\ll}$ d(x), x > $_{({\alpha},{\beta})}$, $x^m$ > $_{({\alpha},{\beta})}=0$, for all $x{\in}R$. Further, if < d(x), x > ${\in}Z(R)$, the center of R, for all $x{\in}R$ or < d(x)x - xd(x), x >= 0, for all $x{\in}R$, then it is proved that d is commuting on R. Some more related results are also obtained for additive mapping on R.

• 충청수학회지
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• 제29권4호
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• pp.523-530
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• 2016

Let R be a 3!-torsion free semiprime ring, and let $D:R{\rightarrow}R$ be a Jordan derivation on a semiprime ring R. In this case, we show that [D(x), x]D(x) = 0 if and only if D(x)[D(x), x] = 0 for every $x{\in}R$. In particular, let A be a Banach algebra with rad(A). If D is a continuous linear Jordan derivation on A, then we see that $[D(x),x]D(x){\in}rad(A)$ if and only if $[D(x),x]D(x){\in}rad(A)$ for all $x{\in}A$.

• 대한수학회논문집
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• 제33권1호
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• pp.103-125
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• 2018

Let R be a 5!-torsion free semiprime ring, and let $D:R{\rightarrow}R$ be a Jordan derivation on a semiprime ring R. Then $[D(x),x]D(x)^2=0$ if and only if $D(x)^2[D(x), x]=0$ for every $x{\in}R$. In particular, let A be a Banach algebra with rad(A) and if D is a continuous linear Jordan derivation on A, then we show that $[D(x),x]D(x)2{\in}rad(A)$ if and only if $D(x)^2[D(x),x]{\in}rad(A)$ for all $x{\in}A$ where rad(A) is the Jacobson radical of A.