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ON RADICALLY-SYMMETRIC IDEALS

  • Hashemi, Ebrahim (Department of Mathematics Shahrood University of Technology)
  • Received : 2009.12.31
  • Published : 2011.07.31

Abstract

A ring R is called symmetric, if abc = 0 implies acb = 0 for a, b, c ${\in}$ R. An ideal I of a ring R is called symmetric (resp. radically-symmetric) if R=I (resp. R/$\sqrt{I}$) is a symmetric ring. We first show that symmetric ideals and ideals which have the insertion of factors property are radically-symmetric. We next show that if R is a semicommutative ring, then $T_n$(R) and R[x]=($x^n$) are radically-symmetric, where ($x^n$) is the ideal of R[x] generated by $x^n$. Also we give some examples of radically-symmetric ideals which are not symmetric. Connections between symmetric ideals of R and related ideals of some ring extensions are also shown. In particular we show that if R is a symmetric (or semicommutative) (${\alpha}$, ${\delta}$)-compatible ring, then R[x; ${\alpha}$, ${\delta}$] is a radically-symmetric ring. As a corollary we obtain a generalization of [13].

Keywords

insertion of factors property;(${\alpha}$, ${\delta}$)-compatible ideals;${\alpha}$-rigid ideals;Ore extensions;symmetric rings;semicommutative rings

References

  1. D. D. Anderson and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra 26 (1998), no. 7, 2265-2272. https://doi.org/10.1080/00927879808826274
  2. E. Hashemi, On ideals which have the weakly insertion of factors property, J. Sci. Islam. Repub. Iran 19 (2008), no. 2, 145-152.
  3. E. Hashemi, Compatible ideals and radicals of Ore extensions, New York J. Math. 12 (2006), 349-356.
  4. E. Hashemi and A. Moussavi, Polynomial extensions of quasi-Baer rings, Acta Math. Hungar. 107 (2005), no. 3, 207-224. https://doi.org/10.1007/s10474-005-0191-1
  5. C. Y. Hong, N. Y. Kim, T. K. Kwak, and Y. Lee, Extensions of zip rings, J. Pure Appl. Algebra 195 (2005), no. 3, 231-242. https://doi.org/10.1016/j.jpaa.2004.08.025
  6. C. Y. Hong, T. K. Kwak, and S. T. Rizvi, Rigid ideals and radicals of Ore extensions, Algebra Colloq. 12 (2005), no. 3, 399-412. https://doi.org/10.1142/S1005386705000374
  7. C. Huh, H. K. Kim, and Y. Lee, P.P.-rings and generalized P.P.-rings, J. Pure Appl. Algebra 167 (2002), no. 1, 37-52. https://doi.org/10.1016/S0022-4049(01)00149-9
  8. C. Huh, Y. Lee, and A. Smoktunowicz, Armendariz rings and semicommutative rings, Comm. Algebra 30 (2002), no. 2, 751-761. https://doi.org/10.1081/AGB-120013179
  9. N. K. Kim and Y. Lee, Armendariz rings and reduced rings, J. Algebra 223 (2000), no. 2, 477-488. https://doi.org/10.1006/jabr.1999.8017
  10. N. K. Kim and Y. Lee, Extensions of reversible rings, J. Pure Appl. Algebra 185 (2003), no. 1-3, 207-223. https://doi.org/10.1016/S0022-4049(03)00109-9
  11. J. Krempa, Some examples of reduced rings, Algebra Colloq. 3 (1996), no. 4, 289-300.
  12. J. Lambek, On the representation of modules by sheaves of factor modules, Canad. Math. Bull. 14 (1971), no. 3, 359-368. https://doi.org/10.4153/CMB-1971-065-1
  13. L. Liang, L.Wang, and Z. Liu, On a generalization of semicommutative rings, Taiwanese J. Math. 11 (2007), no. 5, 1359-1368. https://doi.org/10.11650/twjm/1500404869
  14. G. Mason, Re exive ideals, Comm. Algebra 9 (1981), no. 17, 1709-1724. https://doi.org/10.1080/00927878108822678