DOI QR코드

DOI QR Code

RING WHOSE MAXIMAL ONE-SIDED IDEALS ARE TWO-SIDED

  • Published : 2002.08.01

Abstract

In this note we are concerned with relationships between one-sided ideals and two-sided ideals, and study the properties of polynomial rings whose maximal one-sided ideals are two-sided, in the viewpoint of the Nullstellensatz on noncommutative rings. Let R be a ring and R[x] be the polynomial ring over R with x the indeterminate. We show that eRe is right quasi-duo for $0{\neq}e^2=e{\in}R$ if R is right quasi-duo; R/J(R) is commutative with J(R) the Jacobson radical of R if R[$\chi$] is right quasi-duo, from which we may characterize polynomial rings whose maximal one-sided ideals are two-sided; if R[x] is right quasi-duo then the Jacobson radical of R[x] is N(R)[x] and so the $K\ddot{o}the's$ conjecture (i.e., the upper nilradical contains every nil left ideal) holds, where N(R) is the set of all nilpotent elements in R. Next we prove that if the polynomial rins R[x], over a reduced ring R with $\mid$X$\mid$ $\geq$ 2, is right quasi-duo, then R is commutative. Several counterexamples are included for the situations that occur naturally in the process of this note.

Keywords

quasi-duo ring;polynomial ring;Jacobson radical;commutative ring

References

  1. A. W. Chatters;C. R. Hajarnavis
  2. Memoirs Amer. Math. Soc. v.133 Krull dimension R. Gordon;J. C. Robson
  3. Canad. J. Math. v.21 Nil subrings of Goldie rings are nilpotent C. Lanski https://doi.org/10.4153/CJM-1969-098-x
  4. Comm. Algebra v.27 no.8 On rings in which every maximal one-sided ideal contains a maximal ideal Y. Lee;C. Huh https://doi.org/10.1080/00927879908826676
  5. L. H. Rowen
  6. Comm. Algebra v.23 no.6 On strongly bounded rings and duo rings Y. Hirano;C. Y. Hong;J. Y. Kim;J. K. Park https://doi.org/10.1080/00927879508825341
  7. J. Algebra v.29 no.2 Nil ideals in rings with finite Krull dimension T. H. Lenagan
  8. Bull. Kor. Math. Soc. v.36 no.3 A study on quasi-duo rings C. O. Kim;H. K. Kim;S. H. Jang
  9. American Math. Monthly v.104 no.2 Very semisimple modules W. K. Nicholson https://doi.org/10.2307/2974986
  10. Trans. Amer. Math. Soc. v.95 Finitistic dimension and a generalization of semiprimary rings H. Bass https://doi.org/10.1090/S0002-9947-1960-0157984-8
  11. Glasgow Math. J. v.37 On quasi-duo rings H.-P. Yu https://doi.org/10.1017/S0017089500030342
  12. F. W. Anderson;K. R. Fuller
  13. Pure Appl. Math. Sci. v.21 Weakly right duo rings X. Yao

Cited by

  1. Quasi-duo skew polynomial rings vol.212, pp.8, 2008, https://doi.org/10.1016/j.jpaa.2008.01.002