Ore Extensions over $\sigma$-rigid Rings
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Let $R$ be a ring with an endomorphism $\sigma$ and a $\sigma$-derivation$\delta$. $R$ is called ($\sigma$, $\delta$)-$Baer$(resp. ($\sigma$, $\delta$)-$quasi$-$Baer$, ($\sigma$,$\delta$)-$p.q.$-$Baer$, ($\sigma$, $\delta$)-$p.p.$) if the rightannihilator of every right ($\sigma$, $\delta$)-set (resp.,($\sigma$, $\delta$)-ideal, principal ($\sigma$, $\delta$)-ideal,($\sigma$, $\delta$)-element) of $R$ is generated by an idempotentof $R$. In this paper, for a given Ore extension $A$ = $R[x; \sigma, \delta]$of $R$, the following properties are investigated:If $R$ is a $\sigma$-rigid ringin which $\sigma$ and $\delta$ commute, then(1) $R$ is ($\sigma, \delta)$-Baer if and only if$R$ is ($\sigma, \delta)$-quasi-Baer if and only if $A$ is ($\bar{\sigma},\bar{\delta})$-Baer if and only if $A$ is ($\bar{\sigma},\bar{\delta})$-quasi-Baer;(2) $R$ is ($\sigma, \delta)$-p.p. if and only if$R$ is ($\sigma, \delta)$-$p.q.$-Baer if and only if $A$ is ($\bar{\sigma},\bar{\delta})$-$p.p.$ if and only if $A$ is ($\bar{\sigma},\bar{\delta})$-$p.q.$-Baer.
Abstract
Let $R$ be a ring with an endomorphism $\sigma$ and a $\sigma$-derivation$\delta$. $R$ is called ($\sigma$, $\delta$)-$Baer$(resp. ($\sigma$, $\delta$)-$quasi$-$Baer$, ($\sigma$,$\delta$)-$p.q.$-$Baer$, ($\sigma$, $\delta$)-$p.p.$) if the rightannihilator of every right ($\sigma$, $\delta$)-set (resp.,($\sigma$, $\delta$)-ideal, principal ($\sigma$, $\delta$)-ideal,($\sigma$, $\delta$)-element) of $R$ is generated by an idempotentof $R$. In this paper, for a given Ore extension $A$ = $R[x; \sigma, \delta]$of $R$, the following properties are investigated:If $R$ is a $\sigma$-rigid ringin which $\sigma$ and $\delta$ commute, then(1) $R$ is ($\sigma, \delta)$-Baer if and only if$R$ is ($\sigma, \delta)$-quasi-Baer if and only if $A$ is ($\bar{\sigma},\bar{\delta})$-Baer if and only if $A$ is ($\bar{\sigma},\bar{\delta})$-quasi-Baer;(2) $R$ is ($\sigma, \delta)$-p.p. if and only if$R$ is ($\sigma, \delta)$-$p.q.$-Baer if and only if $A$ is ($\bar{\sigma},\bar{\delta})$-$p.p.$ if and only if $A$ is ($\bar{\sigma},\bar{\delta})$-$p.q.$-Baer.
참고문헌 (20)
- E.P. Armendariz, "A note on extensions of Baer and p.p.-rings", J. Austral. Math. Soc, 18, pp.470~473 (1974)
- H. E. Bell, "Near-rings in which each element is a power of itself", Bull. Austral. Math. Soc, 2, pp.363~368 (1970)
- S.K. Berberian, "Baer-rings" (1972), Springer-Verlag
- J. Bergen, "Skew derivations and the nil and prime radicals", Coll. Math, 128, pp.229~236 (2012)
- G. F. Birkenmeier, "Baer rings and quasi-continuous rings have a MSDN", Pac. J. Math, 97, pp.283~292 (1981)
- G. F. Birkenmeier, "Decompositions of Baer-like rings", Acta Math. Hung, 59, pp.319~326 (1992)
- G. F. Birkenmeier, "On extensions of quasi-Baer and principally quasi-Baer rings", J. of Pure Appl. Algebra, 159, pp.25~42 (2001)
- W. E. Clark, "Twisted matrix units semigroup algebras", Duke Math. J, 34, pp.417~424 (1967)
- K.R. Goodearl, "An introduction to noncommutative Noetherian rings" (2004), Cambridge University Press
- J. Han, "Semiprime ore extensions", Comm. Algebra, 28, pp.3795~3801 (2000)
- C. Y. Hong, "Ore extensions of Baer and p.p.-rings", J. Pure Appl. Algebra, 151, pp.215~226 (2000)
- D. A. Jordan, "Noetherian ore extensions and Jacobson rings", J. London Math. Soc, 10, pp.281~291 (1975)
- I. Kaplansky, "Rings of operators, Math. Lecture Notes Series" (1965), Benjamin
- T.Y. Lam, "Primeness, semiprimess and prime radical of ore extensions", Comm. Algebra, 25, pp.2459~2506 (1997)
- J.C. McConnell, "Noncommutative Noetherian rings" (1987), John Wiley & Sons Ltd
- L. Motais de Narbonne, "Anneaux semi-commutatifs et unis riels anneaux dont les id aux principaux sont idempotents", Proceedings of the 106th National Congress of Learned Societies, pp.71~73 (1982), Bib. Nat
- Alireza R. Nasr-Isfahani, "Skew Laurent polynomial extensions of Baer and p.p.-rings", 대한수학회보, 46(6), pp.1041~1050 (2009), 대한수학회
- P. Pollingher, "On Baer and quasi-Baer rings", Duke Math. J, 37, pp.127~138 (1970)
- L.H. Rowen, "Ring theory" (1988), Academic Press Inc
- G. Shin, "Prime ideals and sheaf representation of a pseudo symmetric ring", Trans. Amer. Math. Soc, 184, pp.43~60 (1973)