• Title, Summary, Keyword: semiprime ring

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SKEW POLYNOMIAL RINGS OVER SEMIPRIME RINGS

  • Hong, Chan-Yong;Kim, Nam-Kyun;Lee, Yang
    • Journal of the Korean Mathematical Society
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    • v.47 no.5
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    • pp.879-897
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    • 2010
  • Y. Hirano introduced the concept of a quasi-Armendariz ring which extends both Armendariz rings and semiprime rings. A ring R is called quasi-Armendariz if $a_iRb_j$ = 0 for each i, j whenever polynomials $f(x)\;=\;\sum_{i=0}^ma_ix^i$, $g(x)\;=\;\sum_{j=0}^mb_jx^j\;{\in}\;R[x]$ satisfy f(x)R[x]g(x) = 0. In this paper, we first extend the quasi-Armendariz property of semiprime rings to the skew polynomial rings, that is, we show that if R is a semiprime ring with an epimorphism $\sigma$, then f(x)R[x; $\sigma$]g(x) = 0 implies $a_iR{\sigma}^{i+k}(b_j)=0$ for any integer k $\geq$ 0 and i, j, where $f(x)\;=\;\sum_{i=0}^ma_ix^i$, $g(x)\;=\;\sum_{j=0}^mb_jx^j\;{\in}\;R[x,\;{\sigma}]$. Moreover, we extend this property to the skew monoid rings, the Ore extensions of several types, and skew power series ring, etc. Next we define $\sigma$-skew quasi-Armendariz rings for an endomorphism $\sigma$ of a ring R. Then we study several extensions of $\sigma$-skew quasi-Armendariz rings which extend known results for quasi-Armendariz rings and $\sigma$-skew Armendariz rings.

ON SUBDIRECT PRODUCT OF PRIME MODULES

  • Dehghani, Najmeh;Vedadi, Mohammad Reza
    • Communications of the Korean Mathematical Society
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    • v.32 no.2
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    • pp.277-285
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    • 2017
  • In the various module generalizations of the concepts of prime (semiprime) for a ring, the question "when are semiprime modules subdirect product of primes?" is a serious question in this context and it is considered by earlier authors in the literature. We continue study on the above question by showing that: If R is Morita equivalent to a right pre-duo ring (e.g., if R is commutative) then weakly compressible R-modules are precisely subdirect products of prime R-modules if and only if dim(R) = 0 and R/N(R) is a semi-Artinian ring if and only if every classical semiprime module is semiprime. In this case, the class of weakly compressible R-modules is an enveloping for Mod-R. Some related conditions are also investigated.

ON PRIME AND SEMIPRIME RINGS WITH SYMMETRIC n-DERIVATIONS

  • Park, Kyoo-Hong
    • 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.

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ON PRIME AND SEMIPRIME RINGS WITH PERMUTING 3-DERIVATIONS

  • Jung, Yong-Soo;Park, Kyoo-Hong
    • 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.

ON 4-PERMUTING 4-DERIVATIONS IN PRIME AND SEMIPRIME RINGS

  • Park, Kyoo-Hong
    • 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.

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ON WEAK ARMENDARIZ RINGS

  • Jeon, Young-Cheol;Kim, Hong-Kee;Lee, Yang;Yoon, Jung-Sook
    • Bulletin of the Korean Mathematical Society
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    • v.46 no.1
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    • pp.135-146
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    • 2009
  • In the present note we study the properties of weak Armendariz rings, and the connections among weak Armendariz rings, Armendariz rings, reduced rings and IFP rings. We prove that a right Ore ring R is weak Armendariz if and only if so is Q, where Q is the classical right quotient ring of R. With the help of this result we can show that a semiprime right Goldie ring R is weak Armendariz if and only if R is Armendariz if and only if R is reduced if and only if R is IFP if and only if Q is a finite direct product of division rings, obtaining a simpler proof of Lee and Wong's result. In the process we construct a semiprime ring extension that is infinite dimensional, from given any semi prime ring. We next find more examples of weak Armendariz rings.

QUASI-ARMENDARIZ PROPERTY FOR SKEW POLYNOMIAL RINGS

  • Baser, Muhittin;Kwa, Tai Keun
    • Communications of the Korean Mathematical Society
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    • v.26 no.4
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    • pp.557-573
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    • 2011
  • The concept of the quasi-Armendariz property of rings properly contains Armendariz rings and semiprime rings. In this paper, we extend the quasi-Armendariz property for a polynomial ring to the skew polynomial ring, hence we call such ring a ${\sigma}$-quasi-Armendariz ring for a ring endomorphism ${\sigma}$, and investigate its structures, several extensions and related properties. In particular, we study the semiprimeness and the quasi-Armendariz property between a ring R and the skew polynomial ring R[x;${\sigma}$$] of R, and so these provide us with an opportunity to study quasi-Armendariz rings and semiprime rings in a general setting, and several known results follow as consequences of our results.

SEMIPRIME NEAR-RINGS WITH ORTHOGONAL DERIVATIONS

  • Park, Kyoo-Hong;Jung, Yong-Soo
    • The Pure and Applied Mathematics
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    • v.13 no.4
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    • pp.303-310
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    • 2006
  • M. $Bre\v{s}ar$ and J. Vukman obtained some results concerning orthogonal derivations in semiprime rings which are related to the result that is well-known to a theorem of Posner for the product of two derivations in prime rings. In this paper, we present orthogonal generalized derivations in semiprime near-rings.

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