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A Survey of Strongly Clean Rings

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Abstract

Let R be an associative ring with identity. An element a∈R is called clean if a=e+u with e an idempotent and u a unit of R and a is called strongly clean if, in addition, eu=ue. A ring R is called clean if every element of R is clean and R is strongly clean if every element of R is strongly clean. In the paper [Nicholson and Zhou, Clean rings: a survey, Advances in Ring Theory, 181–198, World Sci. Pub., Hackensack, NJ, 2005], the authors brought out an up to date account of the results in the study of clean rings. Here, we give an account of the results on strongly clean rings.

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References

  1. Anderson, F.W., Camillo, V.P.: Commutative rings whose elements are a sum of a unit and an idempotent. Commun. Algebra 30, 3327–3336 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  2. Anderson, F.W., Fuller, K.R.: Rings and Categories of Modules, 2nd edn. Springer, New York (1992)

    MATH  Google Scholar 

  3. Ara, P.: Strongly π-regular rings have stable range one. Proc. Am. Math. Soc. 124, 3293–3298 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  4. Arens, R.F., Kaplansky, I.: Topological representation of algebras. Trans. Am. Math. Soc. 63, 457–481 (1948)

    Article  MATH  MathSciNet  Google Scholar 

  5. Armendariz, E.P., Fisher, J.W., Snider, R.L.: On injective and surjective endomorphisms of finitely generated modules. Commun. Algebra 6, 659–672 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  6. Aryapoor, M.: Noncommutative Henselian rings. arXiv:math.RA/0604586

  7. Azarpanah, F.: When is C(X) a clean ring? Acta Math. Hung. 94(1–2), 53–58 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. Azumaya, G.: On maximally central algebras. Nagoya Math. J. 2, 119–150 (1950)

    MathSciNet  Google Scholar 

  9. Azumaya, G.: Strongly π-regular rings. J. Fac. Sci. Hokkaido Univ. 13, 34–39 (1954)

    MathSciNet  Google Scholar 

  10. Borooah, G., Diesl, A.J., Dorsey, T.J.: Strongly clean triangular matrix rings over local rings. J. Algebra 312, 773–797 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Borooah, G., Diesl, A.J., Dorsey, T.J.: Strongly clean matrix rings over commutative local rings. J. Pure Appl. Algebra 212(1), 281–296 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Burgess, W.D., Menal, P.: On strongly π-regular rings and homomorphisms into them. Commun. Algebra 16, 1701–1725 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  13. Camillo, V.P., Khurana, D.: A characterization of unit regular rings. Commun. Algebra 29(5), 2293–2295 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Cedó, F., Rowen, L.H.: Addendum to “Examples of semiperfect rings”, [Israel J. Math. 65 (1989), no. 3, 273–283]. Isr. J. Math. 107, 343–348 (1998)

    Article  MATH  Google Scholar 

  15. Chen, J., Zhou, Y.: Strongly clean power series rings. Proc. Edinb. Math. Soc. 50, 73–85 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  16. Chen, J., Nicholson, W.K., Zhou, Y.: Group rings in which every element is uniquely the sum of a unit and an idempotent. J. Algebra 306, 453–460 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Chen, J., Yang, X., Zhou, Y.: On strongly clean matrix and trianglular matrix rings. Commun. Algebra 34(10), 3659–3674 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  18. Chen, J., Yang, X., Zhou, Y.: When is the 2×2 matrix ring over a commutative local ring strongly clean? J. Algebra 301, 280–293 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  19. Chen, J., Wang, Z., Zhou, Y.: Rings in which elements are uniquely the sum of an idempotent and a unit that commute. J. Pure Appl. Algebra 213(2), 215–223 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  20. Chen, W.: A question on strongly clean rings. Commun. Algebra 34(7), 2374–2350 (2006)

    Google Scholar 

  21. Cohn, P.M.: Free Rings and Their Relations, 2nd edn. Academic, New York (1985)

    MATH  Google Scholar 

  22. Couchot, F.: Strong cleanness of matrix rings over commutative rings. Commun. Algebra 36(2), 346–351 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  23. Dischinger, F.: Sur les anneaus fortement π-régliers. C.R. Acad. Sci. Paris 283, 571–573 (1976)

    MATH  MathSciNet  Google Scholar 

  24. Ehrlich, G.: Unit regular rings. Portugal. Math. 27, 209–212 (1968)

    MATH  MathSciNet  Google Scholar 

  25. Eisenbud, D.: Commutative Algebra With a View Toward Algebraic Geometry. Graduate Texts in Mathematics, vol. 150. Springer, Berlin (1995)

    MATH  Google Scholar 

  26. Engelking, R.: General Topology. Heldermann Verlag, Berlin (1989)

    MATH  Google Scholar 

  27. Gillman, L., Jerson, M.: Rings of Continuous Functions. Springer, New York (1976)

    MATH  Google Scholar 

  28. Fan, L., Yang, X.: On strongly clean matrix rings. Glasg. Math. J. 45(3), 557–566 (2006)

    Article  MathSciNet  Google Scholar 

  29. Han, J., Nicholson, W.K.: Extensions of clean rings. Commun. Algebra 29, 2589–2595 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  30. Hirano, Y.: Some characterizations of π-regular rings of bounded index. Math. J. Okayama Univ. 32, 97–101 (1990)

    MATH  MathSciNet  Google Scholar 

  31. Kaplansky, I.: Topological representation of algebras (II). Trans. Am. Math. Soc. 68, 62–75 (1950)

    Article  MATH  MathSciNet  Google Scholar 

  32. Li, Y.: On strongly clean matrix rings. J. Algebra 312(1), 397–404 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  33. McCoy, N.: Generalized regular rings. Bull. Am. Math. Soc. 45, 175–178 (1939)

    Article  MathSciNet  Google Scholar 

  34. McGovern, W.W.: Clean semiprime f-rings with bounded inversion. Commun. Algebra 31(7), 3295–3304 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  35. Nagata, M.: On the theory of Henselian rings. Nagoya Math. J. 5, 45–57 (1953)

    MATH  MathSciNet  Google Scholar 

  36. Nicholson, W.K.: Lifting idempotents and exchange rings. Trans. Am. Math. Soc. 229, 269–278 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  37. Nicholson, W.K.: Strongly clean rings and Fitting’s lemma. Commun. Algebra 27, 3583–3592 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  38. Nicholson, W.K., Zhou, Y.: Clean rings: a survey. In: Advances in Ring Theory, pp. 181–198. World Sci. Pub., Hackensack (2005)

    Google Scholar 

  39. Parshin, A.N.: On a ring of formal pseudo-differential operators. Proc. Steklov Inst. Math. 224(1), 266–280 (1999)

    MathSciNet  Google Scholar 

  40. Sánchez Campos, E.: On strongly clean rings. Unpublished (2002)

  41. von Neumann, J.: Regular rings. Proc. Natl. Acad. Sci. U.S.A. 22, 707–713 (1936)

    Article  Google Scholar 

  42. Wang, Z., Chen, J.: On two open problems about strongly clean rings. Bull. Aust. Math. Soc. 70, 279–282 (2004)

    Article  MATH  Google Scholar 

  43. Yang, X.: Strongly clean rings and g(x)-clean rings. Ph.D. thesis, Memorial University of Newfoundland (2007)

  44. Yang, X., Zhou, Y.: Some families of strongly clean rings. Linear Algebra Appl. 425, 119–129 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  45. Yang, X., Zhou, Y.: Strongly clean property of the 2×2 matrix ring over a general local ring. J. Algebra 320, 2280–2290 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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  1. Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

    Xiande Yang

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  1. Xiande Yang
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Yang, X. A Survey of Strongly Clean Rings. Acta Appl Math 108, 157–173 (2009). https://doi.org/10.1007/s10440-008-9364-6

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  • DOI: https://doi.org/10.1007/s10440-008-9364-6

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