Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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ON LIFTING OF STABLE RANGE ONE ELEMENTS

  • Received : 2019.06.01
  • Accepted : 2019.08.14
  • Published : 2020.05.01
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Abstract

Stable range of rings is a unifying concept for problems related to the substitution and cancellation of modules. The newly appeared element-wise setting for the simplest case of stable range one is tempting to study the lifting property modulo ideals. We study the lifting of elements having (idempotent) stable range one from a quotient of a ring R modulo a two-sided ideal I by providing several examples and investigating the relations with other lifting properties, including lifting idempotents, lifting units, and lifting of von Neumann regular elements. In the case where the ring R is a left or a right duo ring, we show that stable range one elements lift modulo every two-sided ideal if and only if R is a ring with stable range one. Under a mild assumption, we further prove that the lifting of elements having idempotent stable range one implies the lifting of von Neumann regular elements.

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References

  1. F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, second edition, Graduate Texts in Mathematics, 13, Springer-Verlag, New York, 1992. https://doi.org/10.1007/978-1-4612-4418-9
  2. G. Baccella, Semi-Artinian V-rings and semi-Artinian von Neumann regular rings, J. Algebra 173 (1995), no. 3, 587-612. https://doi.org/10.1006/jabr.1995.1104
  3. G. Baccella, Exchange property and the natural preorder between simple modules over semi-Artinian rings, J. Algebra 253 (2002), no. 1, 133-166. https://doi.org/10.1016/S0021-8693(02)00044-3
  4. H. Bass, K-theory and stable algebra, Inst. Hautes Etudes Sci. Publ. Math. No. 22 (1964), 5-60.
  5. H. Chen, Rings with many idempotents, Int. J. Math. Math. Sci. 22 (1999), no. 3, 547-558. https://doi.org/10.1155/S0161171299225471
  6. D. Estes and J. Ohm, Stable range in commutative rings, J. Algebra 7 (1967), 343-362. https://doi.org/10.1016/0021-8693(67)90075-0
  7. M. A. Fortes Escalona, I. de las Penas Cabrera, and E. Sanchez Campos, Lifting idempotents in associative pairs, J. Algebra 222 (1999), no. 2, 511-523. https://doi.org/10.1006/jabr.1999.8025
  8. L. Fuchs, On a substitution property of modules, Monatsh. Math. 75 (1971), 198-204. https://doi.org/10.1007/BF01299099
  9. S. Garg, H. K. Grover, and D. Khurana, Perspective rings, J. Algebra 415 (2014), 1-12. https://doi.org/10.1016/j.jalgebra.2013.09.055
  10. K. R. Goodearl, von Neumann Regular Rings, second edition, Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1991.
  11. M. Henriksen, On a class of regular rings that are elementary divisor rings, Arch. Math. (Basel) 24 (1973), 133-141. https://doi.org/10.1007/BF01228189
  12. V. A. Hiremath and S. Hegde, Using ideals to provide a unified approach to uniquely clean rings, J. Aust. Math. Soc. 96 (2014), no. 2, 258-274. https://doi.org/10.1017/S1446788713000591
  13. I. Kaplansky, Bass's first stable range condition, mimeographed notes, 1971.
  14. D. Khurana and T. Y. Lam, Clean matrices and unit-regular matrices, J. Algebra 280 (2004), no. 2, 683-698. https://doi.org/10.1016/j.jalgebra.2004.04.019
  15. D. Khurana and T. Y. Lam, Rings with internal cancellation, J. Algebra 284 (2005), no. 1, 203-235. https://doi.org/10.1016/j.jalgebra.2004.07.032
  16. D. Khurana, T. Y. Lam, and P. P. Nielsen, An ensemble of idempotent lifting hypotheses, J. Pure Appl. Algebra 222 (2018), no. 6, 1489-1511. https://doi.org/10.1016/j.jpaa.2017.07.008
  17. T. Y. Lam, A crash course on stable range, cancellation, substitution, and exchange, J. Algebra Appl. 3 (2004), 301-343. https://doi.org/10.1142/S0219498804000897
  18. P. Menal and J. Moncasi, On regular rings with stable range 2, J. Pure Appl. Algebra 24 (1982), no. 1, 25-40. https://doi.org/10.1016/0022-4049(82)90056-1
  19. P. Menal and J. Moncasi, Lifting units in self-injective rings and an index theory for Rickart $C^\ast$-algebras, Pacific J. Math. 126 (1987), no. 2, 295-329. http://projecteuclid.org/euclid.pjm/1102699806 https://doi.org/10.2140/pjm.1987.126.295
  20. W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977), 269-278. https://doi.org/10.2307/1998510
  21. A. C. Ozcan, A. Harmanci, and P. F. Smith, Duo modules, Glasg. Math. J. 48 (2006), no. 3, 533-545. https://doi.org/10.1017/S0017089506003260
  22. F. Perera, Lifting units modulo exchange ideals and C*-algebras with real rank zero, J. Reine Angew. Math. 522 (2000), 51-62. https://doi.org/10.1515/crll.2000.040
  23. F. Siddique, On two questions of Nicholson, https://arxiv.org/pdf/1402.4706.pdf, (2014), 5 pages.
  24. J. Ster, Lifting units in clean rings, J. Algebra 381 (2013), 200-208. https://doi.org/10.1016/j.jalgebra.2013.02.014
  25. L. N. Vaserstein, The stable range of rings and the dimension of topological spaces, Funkcional. Anal. i Prilozen. 5 (1971), no. 2, 17-27.
  26. L. N. Vaserstein, Bass's first stable range condition, J. Pure Appl. Algebra 34 (1984), no. 2-3, 319-330. https://doi.org/10.1016/0022-4049(84)90044-6
  27. Z.Wang, J. Chen, D. Khurana, and T.Y. Lam, Rings of idempotent stable range one, Algebr. Represent. Theory 15 (2012), no. 1, 195-200. https://doi.org/10.1007/s10468-011-9276-4
  28. C. A. Weibel, The K-book, Graduate Studies in Mathematics, 145, American Mathematical Society, Providence, RI, 2013.
  29. H.-P. Yu, On quasi-duo rings, Glasgow Math. J. 37 (1995), no. 1, 21-31. https://doi.org/10.1017/S0017089500030342
  30. Y. Zhou, Generalizations of perfect, semiperfect, and semiregular rings, Algebra Colloq. 7 (2000), no. 3, 305-318. https://doi.org/10.1007/s10011-000-0305-9