Communications of the Korean Mathematical Society (대한수학회논문집)

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THE COHN-JORDAN EXTENSION AND SKEW MONOID RINGS OVER A QUASI-BAER RING

  • HASHEMI EBRAHIM (Department of Mathematics Shahrood University of Technology)
  • Published : 2006.01.01
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Abstract

A ring R is called (left principally) quasi-Baer if the left annihilator of every (principal) left ideal of R is generated by an idempotent. Let R be a ring, G be an ordered monoid acting on R by $\beta$ and R be G-compatible. It is shown that R is (left principally) quasi-Baer if and only if skew monoid ring $R_{\beta}[G]$ is (left principally) quasi-Baer. If G is an abelian monoid, then R is (left principally) quasi-Baer if and only if the Cohn-Jordan extension $A(R,\;\beta)$ is (left principally) quasi-Baer if and only if left Ore quotient ring $G^{-1}R_{\beta}[G]$ is (left principally) quasi-Baer.

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References

  1. E. P. Armendariz, A note on extensions of Baer and p.p-rings, J. Austral. Math. Soc. 18 (1974), 470-473 https://doi.org/10.1017/S1446788700029190
  2. G. F. Birkenmeier, Idempotents and completely semiprime ideals, Comm. Algebra 11 (1983), 567-580 https://doi.org/10.1080/00927878308822865
  3. G. F. Birkenmeier, On polynomial extensions of principally quasi-Baer rings, Kyungpook Math. J. 40 (2000), 247-253
  4. G. F. Birkenmeier, J. Y. Kim, and J. K. Park, Principally quasi-Baer rings, Comm. Algebra 29 (2001), no. 2, 639-660 https://doi.org/10.1081/AGB-100001530
  5. G. F. Birkenmeier, J. Y. Kim, and J. K. Park, Polynomial extensions of Baer and quasi-Baer rings, J. Pure Appl. Al- gebra 159 (2001), 24-42
  6. G. F. Birkenmeier, J. Y. Kim, and J. K. Park, On quasi-Baer rings, Contemp. Math. 259 (2000), 67-92 https://doi.org/10.1090/conm/259/04088
  7. W. E. Clark, Twisted matrix units semigroup algebras, Duke Math. J. 34 (1967), 417-424 https://doi.org/10.1215/S0012-7094-67-03446-1
  8. A. W. Chattters and C. R. Hajarnavis, Rings with chain conditions, Pitman, Boston, 1980
  9. J. A. Fraser and W. K. Nicholson, Reduced PP-rings, Math. Japonica 34 (1989), no. 5, 715-725
  10. J. Han, Y. Hirano, and H. Kim, Some results on skew polynomial rings over reduced rings, preprint
  11. 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
  12. E. Hashemi and A. Moussavi, Skew power series extentions of ${\alpha}$-rigid p.p.-rings, Bull. Korean Math. Soc. 41 (2004), no. 4, 657-665 https://doi.org/10.4134/BKMS.2004.41.4.657
  13. E. Hashemi and A. Moussavi, On (${\alpha},{\delta}$)-skew Armendariz Rings, J. Korean Math. Soc. 42 (2005), no. 2, 353-363 https://doi.org/10.4134/JKMS.2005.42.2.353
  14. E. Hashemi, A. Moussavi, and H. S. Javadi, Polynomial ore extentions of Baer and p.p.-rings, Bull. Iranian Math. Soc. 29 (2003), no. 2, 65-84
  15. Y. Hirano, On annihilator ideals of a polynomial ring over a noncommutative ring, J. Pure Appl. Algebra 168 (2002), 45-52 https://doi.org/10.1016/S0022-4049(01)00053-6
  16. Y. Hirano, On ordered monoid rings over a quasi-Baer ring, Comm. Algebra 29 (2001), no. 5, 2089-2095 https://doi.org/10.1081/AGB-100002171
  17. C. Y. Hong, N. K. Kim, and T. K. Kwak, Ore extensions of Baer and p.p-rings, J. Pure Appl. Algebra 151 (2000), 215-226 https://doi.org/10.1016/S0022-4049(99)00020-1
  18. D. A. Jordan, Bijective extension of injective ring endomorphisms, J. London Math. Soc. 35 (1982), no. 2, 435-488
  19. I. Kaplansky, Rings of Operators, Benjamin, New York, 1965
  20. N. H. Kim and Y. Lee, Armendariz rings and reduced rings, J. Algebra 223 (2000), 477-488 https://doi.org/10.1006/jabr.1999.8017
  21. J. Krempa, Some examples of reduced rings, Algebra Colloq. 3 (1996), no. 4, 289-300
  22. T. Y. Lam, A. Leroy, and J. Matczuk, Primeness, semiprimeness and prime radical of ore extensions, Comm. Algebra 28 (2000), no. 8, 3895-3801
  23. Z. Liu, A note on principally quasi-Baer rings, Comm. Algebra 30 (2002), no. 8, 3885-3890 https://doi.org/10.1081/AGB-120005825
  24. A. Moussavi, H. S. Javadi, and E. Hashemi, Generalized quasi-Baer rings, to appear in the Comm. Algebra
  25. A. Moussavi and E. Hashemi, Semiprime skew polynomial rings, submitted
  26. V. A. Mushrub, On the uniform dimension of skew polynomial rings in many variables, Fundamental and applied Mathematics 7 (2001), no. 4, 1107-1121
  27. D. S. Passman, The Algebraic Structure of Group Rings, John Wiley and Sons., 1977
  28. M. B. Rege and S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), 14-17
  29. B. Stenstrom, Rings of Quotients, Springer, Berlin, 1975
  30. H. Tominaga, On s-unital rings, Math. J. Okayama Univ. 18 (1976), 117-134

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