• Title/Summary/Keyword: Semiprime-rings

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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 s.34
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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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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.

GENERALIZED DERIVATIONS AND DERIVATIONS OF RINGS AND BANACH ALGEBRAS

  • Jung, Yong-Soo
    • Honam Mathematical Journal
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    • v.35 no.4
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    • pp.625-637
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    • 2013
  • We investigate anti-centralizing and skew-centralizing mappings involving generalized derivations and derivations on prime and semiprime rings. We also obtain some range inclusion results for generalized linear derivations and linear derivations on Banach algebras by applying the algebraic techniques. Some results in this note are to improve the ones in [22].

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.

SOME CONDITIONS ON DERIVATIONS IN PRIME NEAR-RINGS

  • Cho, Yong-Uk
    • The Pure and Applied Mathematics
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    • v.8 no.2
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    • pp.145-152
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    • 2001
  • Posner [Proc. Amer. Math. Soc. 8 (1957), 1093-1100] defined a derivation on prime rings and Herstein [Canad, Math. Bull. 21 (1978), 369-370] derived commutative property of prime ring with derivations. Recently, Bergen [Canad. Math. Bull. 26 (1983), 267-227], Bell and Daif [Acta. Math. Hunger. 66 (1995), 337-343] studied derivations in primes and semiprime rings. Also, in near-ring theory, Bell and Mason [Near-Rungs and Near-Fields (pp. 31-35), Proceedings of the conference held at the University of Tubingen, 1985. Noth-Holland, Amsterdam, 1987; Math. J. Okayama Univ. 34 (1992), 135-144] and Cho [Pusan Kyongnam Math. J. 12 (1996), no. 1, 63-69] researched derivations in prime and semiprime near-rings. In this paper, Posner, Bell and Mason's results are extended in prime near-rings with some conditions.

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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.

A NOTE ON MULTIPLICATIVE (GENERALIZED)-DERIVATION IN SEMIPRIME RINGS

  • REHMAN, NADEEM UR;HONGAN, MOTOSHI
    • Journal of applied mathematics & informatics
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    • v.36 no.1_2
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    • pp.81-92
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    • 2018
  • In this article we study two Multiplicative (generalized)- derivations ${\mathcal{G}}$ and ${\mathcal{H}}$ that satisfying certain conditions in semiprime rings and tried to find out some information about the associated maps. Moreover, an example is given to demonstrate that the semiprimeness imposed on the hypothesis of the various results is essential.