• Title, Summary, Keyword: sigma-prime ring

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PRIME RADICALS OF SKEW LAURENT POLYNOMIAL RINGS

  • Han, Jun-Cheol
    • Bulletin of the Korean Mathematical Society
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    • v.42 no.3
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    • pp.477-484
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    • 2005
  • Let R be a ring with an automorphism 17. An ideal [ of R is ($\sigma$-ideal of R if $\sigma$(I).= I. A proper ideal P of R is ($\sigma$-prime ideal of R if P is a $\sigma$-ideal of R and for $\sigma$-ideals I and J of R, IJ $\subseteq$ P implies that I $\subseteq$ P or J $\subseteq$ P. A proper ideal Q of R is $\sigma$-semiprime ideal of Q if Q is a $\sigma$-ideal and for a $\sigma$-ideal I of R, I$^{2}$ $\subseteq$ Q implies that I $\subseteq$ Q. The $\sigma$-prime radical is defined by the intersection of all $\sigma$-prime ideals of R and is denoted by P$_{(R). In this paper, the following results are obtained: (1) For a principal ideal domain R, P$_{(R) is the smallest $\sigma$-semiprime ideal of R; (2) For any ring R with an automorphism $\sigma$ and for a skew Laurent polynomial ring R[x, x$^{-1}$; $\sigma$], the prime radical of R[x, x$^{-1}$; $\sigma$] is equal to P$_{(R)[x, x$^{-1}$; $\sigma$ ].

PRIME RADICALS IN ORE EXTENSIONS

  • Han, Jun-Cheol
    • East Asian mathematical journal
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    • v.18 no.2
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    • pp.271-282
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    • 2002
  • Let R be a ring with an endomorphism $\sigma$ and a derivation $\delta$. An ideal I of R is ($\sigma,\;\delta$)-ideal of R if $\sigma(I){\subseteq}I$ and $\delta(I){\subseteq}I$. An ideal P of R is a ($\sigma,\;\delta$)-prime ideal of R if P(${\neq}R$) is a ($\sigma,\;\delta$)-ideal and for ($\sigma,\;\delta$)-ideals I and J of R, $IJ{\subseteq}P$ implies that $I{\subseteq}P$ or $J{\subseteq}P$. An ideal Q of R is ($\sigma,\;\delta$)-semiprime ideal of R if Q is a ($\sigma,\;\delta$)-ideal and for ($\sigma,\;\delta$)-ideal I of R, $I^2{\subseteq}Q$ implies that $I{\subseteq}Q$. The ($\sigma,\;\delta$)-prime radical (resp. prime radical) is defined by the intersection of all ($\sigma,\;\delta$)-prime ideals (resp. prime ideals) of R and is denoted by $P_{(\sigma,\delta)}(R)$(resp. P(R)). In this paper, the following results are obtained: (1) $P_{(\sigma,\delta)}(R)$ is the smallest ($\sigma,\;\delta$)-semiprime ideal of R; (2) For every extended endomorphism $\bar{\sigma}$ of $\sigma$, the $\bar{\sigma}$-prime radical of an Ore extension $P(R[x;\sigma,\delta])$ is equal to $P_{\sigma,\delta}(R)[x;\sigma,\delta]$.

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ON GENERALIZED LIE IDEALS IN SEMI-PRIME RINGS WITH DERIVATION

  • Ozturk, M. Ali;Ceven, Yilmaz
    • East Asian mathematical journal
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    • v.21 no.1
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    • pp.1-7
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    • 2005
  • The object of this paper is to study($\sigma,\;\tau$)-Lie ideals in semi-prime rings with derivation. Main result is the following theorem: Let R be a semi-prime ring with 2-torsion free, $\sigma$ and $\tau$ two automorphisms of R such that $\sigma\tau=\tau\sigma$=, U be both a non-zero ($\sigma,\;\tau$)-Lie ideal and subring of R. If $d^2(U)=0$, then d(U)=0 where d a non-zero derivation of R such that $d\sigma={\sigma}d,\;d\tau={\tau}d$.

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On Prime Near-rings with Generalized (σ,τ)-derivations

  • Golbasi, Oznur
    • Kyungpook Mathematical Journal
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    • v.45 no.2
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    • pp.249-254
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    • 2005
  • Let N be a prime left near-ring with multiplicative center Z and f be a generalized $({\sigma},{\tau})-derivation$ associated with d. We prove commutativity theorems in prime near- rings with generalized $({\sigma},{\tau})-derivation$.

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On Commutativity of σ-Prime Γ-Rings

  • DEY, KALYAN KUMAR;PAUL, AKHIL CHANDRA;DAVVAZ, BIJAN
    • Kyungpook Mathematical Journal
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    • v.55 no.4
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    • pp.827-835
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    • 2015
  • Let U be a ${\sigma}$-square closed Lie ideal of a 2-torsion free ${\sigma}$-prime ${\Gamma}$-ring M. Let $d{\neq}1$ be an automorphism of M such that $[u,d(u)]_{\alpha}{\in}Z(M)$ on U, $d{\sigma}={\sigma}d$ on U, and there exists $u_0$ in $Sa_{\sigma}(M)$ with $M{\Gamma}u_0{\subseteq}U$. Then, $U{\subseteq}Z(M)$. By applying this result, we generalize the results of Oukhtite and Salhi respect to ${\Gamma}$-rings. Finally, for a non-zero derivation of a 2-torsion free ${\sigma}$-prime $\Gamma$-ring, we obtain suitable conditions under which the $\Gamma$-ring must be commutative.

SOME RESULTS CONCERNING ($\theta,\;\varphi$)-DERIVATIONS ON PRIME RINGS

  • Park, Kyoo-Hong;Jung Yong-Soo
    • The Pure and Applied Mathematics
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    • v.10 no.4
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    • pp.207-215
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    • 2003
  • Let R be a prime ring with characteristic different from two and let $\theta,\varphi,\sigma,\tau$ be the automorphisms of R. Let d : $R{\rightarrow}R$ be a nonzero ($\theta,\varphi$)-derivation. We prove the following results: (i) if $a{\in}R$ and [d(R), a]$_{{\theta}o{\sigma},{\varphi}o{\tau}}$=0, then $\sigma(a)\;+\;\tau(a)\;\in\;Z$, the center of R, (ii) if $d([R,a]_{\sigma,\;\tau)\;=\;0,\;then\;\sigma(a)\;+\;\tau(a)\;\in\;Z$, (iii) if $[ad(x),\;x]_{\sigma,\;\tau}\;=\;0;for\;all\;x\;\in\;RE$, then a = 0 or R is commutative.

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SOME RESULTS ON GENERALIZED LIE IDEALS WITH DERIVATION

  • Aydin, Neset;Kaya, Kazim;Golbasi, Oznur
    • East Asian mathematical journal
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    • v.17 no.2
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    • pp.225-232
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    • 2001
  • Let R be a prime ring with characteristic not two. U a (${\sigma},{\tau}$)-left Lie ideal of R and d : R$\rightarrow$R a non-zero derivation. The purpose of this paper is to invesitigate identities satisfied on prime rings. We prove the following results: (1) [d(R),a]=0$\Leftrightarrow$d([R,a])=0. (2) if $(R,a)_{{\sigma},{\tau}}$=0 then $a{\in}Z$. (3) if $(R,a)_{{\sigma},{\tau}}{\subset}C_{{\sigma},{\tau}}$ then $a{\in}Z$. (4) if $(U,a){\subset}Z$ then $a^2{\in}Z\;or\;{\sigma}(u)+{\tau}(u){\in}Z$, for all $u{\in}U$. (5) if $(U,R)_{{\sigma},{\tau}}{\subset}C_{{\sigma},{\tau}}$ then $U{\subset}Z$.

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LIE IDEALS AND DERIVATIONS OF $\sigma$-PRIME RINGS

  • Shuliang, Huang
    • The Pure and Applied Mathematics
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    • v.17 no.1
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    • pp.87-92
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    • 2010
  • Let R be a 2-torsion free $\sigma$-prime ring with an involution $\sigma$, U a nonzero square closed $\sigma$-Lie ideal, Z(R) the center of Rand d a derivation of R. In this paper, it is proved that d = 0 or $U\;{\subseteq}\;Z(R)$ if one of the following conditions holds: (1) $d(xy)\;-\;xy\;{\in}\;Z(R)$ or $d(xy)\;-\;yx\;{\in}Z(R)$ for all x, $y\;{\in}\;U$. (2) $d(x)\;{\circ}\;d(y)\;=\;0$ or $d(x)\;{\circ}\;d(y)\;=\;x\;{\circ}\;y$ for all x, $y\;{\in}\;U$ and d commutes with $\sigma$.

SOME RESULTS ON ENDOMORPHISMS OF PRIME RING WHICH ARE $(\sigma,\tau)$-DERIVATION

  • Golbasi, Oznur;Aydin, Neset
    • East Asian mathematical journal
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    • v.18 no.2
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    • pp.195-203
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    • 2002
  • Let R be a prime ring with characteristic not two and U is a nonzero left ideal of R which contains no nonzero nilpotent right ideal as a ring. For a $(\sigma,\tau)$-derivation d : R$\rightarrow$R, we prove the following results: (1) If d is an endomorphism on R then d=0. (2) If d is an anti-endomorphism on R then d=0. (3) If d(xy)=d(yx), for all x, y$\in$R then R is commutative. (4) If d is an homomorphism or anti-homomorphism on U then d=0.

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