Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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RINGS WITH MANY REGULAR ELEMENTS

  • Received : 2016.05.04
  • Published : 2017.04.30
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Abstract

In this paper we introduce rings that satisfy regular 1-stable range. These rings are left-right symmetric and are generalizations of unit 1-stable range. We investigate characterizations of these kind of rings and show that these rings are closed under matrix rings and Morita Context rings.

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References

  1. P. Ara, Strongly-regular rings have stable range one, Proc. Amer. Math. Soc. 124 (1996), no. 11, 3293-3298. https://doi.org/10.1090/S0002-9939-96-03473-9
  2. N. Ashrafi and E. Nasibi, r-clean rings, Math. Rep. (Bucur.) 15 (65) (2013), no. 2, 125-132.
  3. N. Ashrafi and E. Nasibi, Rings in which elements are the sum of an idempotent and a regular element, Bull. Iranian Math. Soc. 39 (2013), no. 3, 579-588.
  4. V. P. Camillo and H. P. Yu, Stable range one for rings with many idempotents, Trans. Amer. Math. Soc. 347 (1995), no. 8, 3141-3147. https://doi.org/10.1090/S0002-9947-1995-1277100-2
  5. M. C. Campany, M. G. Lozano, and M. S. Molina, Exchange Morita rings, Comm. Algebra 29 (2001), no. 2, 907-925. https://doi.org/10.1081/AGB-100001551
  6. H. Chen, Rings with many idempotents, Int. J. Math. Math. Sci. 22 (1999), no. 3, 547-558. https://doi.org/10.1155/S0161171299225471
  7. H. Chen, Exchange rings satisfying unit 1-stable range, Kyushu J. Math. 54 (2000), no. 1, 1-6. https://doi.org/10.2206/kyushujm.54.1
  8. H. Chen, On stable range conditions, Comm. Algebra 28 (2000), no. 8, 3913-3924. https://doi.org/10.1080/00927870008827065
  9. H. Chen, Units, idempotents, and stable range conditions, Comm. Algebra 29 (2001), no. 2, 703-717. https://doi.org/10.1081/AGB-100001535
  10. H. Chen, Morita contexts with many units, Comm. Algebra 30 (2002), no. 3, 1499-1512. https://doi.org/10.1080/00927870209342393
  11. H. Chen, Exchange rings having stable range one, Chinese J. Contemp. Math. 29 (2008), no. 1, 95-102.
  12. H. Chen and M. Chen, On unit 1-stable range, J. Appl. Algebra Discrete Struct. 1 (2003), no. 3, 189-196.
  13. H. Chen and M. Chen, Unit 1-stable range for ideals, Int. J. Math. Math. Sci. 2004 (2004), no. 45-48, 2477-2482. https://doi.org/10.1155/S0161171204402361
  14. H. Chen and F. Li, Rings with many unit-regular elements, Chinese J. Contemp. Math. 21 (2000), no. 1, 33-38.
  15. E. G. Evans, Krull-Schmidt and cancellation over local rings, Pacifica J. Math. 46 (1973), 115-121. https://doi.org/10.2140/pjm.1973.46.115
  16. K. R. Goodearl and P. Mental, Stable range one for rings with many units, J. Pure Appl. Algebra 54 (1988), no. 2-3, 261-287. https://doi.org/10.1016/0022-4049(88)90034-5
  17. A. Haghany, Hopfcity and co-Hopfcity for Morita contexts, Comm. Algebra 27 (1999), no. 1, 477-492. https://doi.org/10.1080/00927879908826443
  18. W. K. Nicholson, Lifting idempotent and exchange rings, Trans. Amer. Math. Soc. 229 (1977), 269-278. https://doi.org/10.1090/S0002-9947-1977-0439876-2
  19. W. K. Nicholson, Strongly clean rings and fittings lemma, Comm. Algebra 27 (1999), no. 8, 3583-3592. https://doi.org/10.1080/00927879908826649
  20. 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
  21. R. B. Warfield, Jr., Cancellation of modules and groups and stable range of endomor-phism rings, Pacific J. Math. 91 (1980), no. 2, 457-485. https://doi.org/10.2140/pjm.1980.91.457
  22. G. Xiao and W. Tong, n-clean rings and weakly unit stable range rings, Comm. Algebra 33 (2005), no. 5, 1501-1517. https://doi.org/10.1081/AGB-200060531
  23. H. P. Yu, Stable range one for exchange rings, J. Pure Appl. Algebra 98 (1995), no. 1, 105-109. https://doi.org/10.1016/0022-4049(95)90029-2