• 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 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 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 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.

FULLY PRIME MODULES AND FULLY SEMIPRIME MODULES

  • Beachy, John A.;Medina-Barcenas, Mauricio
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.5
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    • pp.1177-1193
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    • 2020
  • Fully prime rings (in which every proper ideal is prime) have been studied by Blair and Tsutsui, and fully semiprime rings (in which every proper ideal is semiprime) have been studied by Courter. For a given module M, we introduce the notions of a fully prime module and a fully semiprime module, and extend certain results of Blair, Tsutsui, and Courter to the category subgenerated by M. We also consider the relationship between the conditions (1) M is a fully prime (semiprime) module, and (2) the endomorphism ring of M is a fully prime (semiprime) ring.

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.