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Strong separativity over exchange rings

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Abstract

An exchange ring R is strongly separative provided that for all finitely generated projective right R-modules A and B, A ⊕ A ≅ A ⊕ B ⇒ A ≅ B. We prove that an exchange ring R is strongly separative if and only if for any corner S of R, aS + bS = S implies that there exist u, v ∈ S such that au = bv and Su + Sv = S if and only if for any corner S of R, aS + bS = S implies that there exists a right invertible matrix \( \left( {\begin{array}{*{20}c} a & b \\ * & * \\ \end{array} } \right) \) ∈ M 2(S). The dual assertions are also proved.

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References

  1. P. Ara: Extensions of exchange rings. J. Algebra 197 (1997), 409–423.

    Article  MATH  MathSciNet  Google Scholar 

  2. P. Ara, K. R. Goodearl, K. C. O’Meara, E. Pardo: Diagonalization of matrices over regular rings. Linear Algebra Appl. 265 (1997), 147–163.

    Article  MATH  MathSciNet  Google Scholar 

  3. P. Ara, K. R. Goodearl, K. C. O’Meara, E. Pardo: Separative cancellation for projective modules over exchange rings. Isr. J. Math. 105 (1998), 105–137.

    Article  MATH  MathSciNet  Google Scholar 

  4. P. Ara, K. C. O’Meara, D. V. Tyukavkin: Cancellation of projective modules over regular rings with comparability. J. Pure Appl. Algebra 107 (1996), 19–38.

    Article  MATH  MathSciNet  Google Scholar 

  5. G. Brookfield: Direct sum cancellation of Noetherian modules. J. Algebra 200 (1998), 207–224.

    Article  MATH  MathSciNet  Google Scholar 

  6. H. Chen: Exchange rings over which every regular matrix admits diagonal reduction. J. Algebra Appl. 3 (2004), 207–217.

    Article  MATH  MathSciNet  Google Scholar 

  7. K. R. Goodearl: Von Neumann Regular Rings. Pitman, London-San Francisco-Melbourne, 1979; Krieger, Malabar (2nd edition), 1991.

    MATH  Google Scholar 

  8. K. R. Goodearl: Von Neumann regular rings and direct sum decomposition problems. Abelian Groups and Modules. Kluwer, Dordrecht, 1995, pp. 249–255.

    Google Scholar 

  9. P. Menal, J. Moncasi: On regular rings with stable range 2. J. Pure Appl. Algebra 24 (1982), 25–40.

    Article  MATH  MathSciNet  Google Scholar 

  10. W. K. Nicholson: On exchange rings. Commun. Algebra 25 (1997), 1917–1918.

    Article  MATH  MathSciNet  Google Scholar 

  11. K. C. O’Meara, C. Vinsonhaler: Separative cancellation and multiple isomorphism in torsion-free Abelian groups. J. Algebra 221 (1999), 536–550.

    Article  MATH  MathSciNet  Google Scholar 

  12. E. Pardo: Comparability, separativity and exchange rings. Commun. Algebra 24 (1996), 2915–2929.

    Article  MATH  MathSciNet  Google Scholar 

  13. A. A. Tuganbaev: Rings Close to Regular. Kluwer, Dordrecht, 2002.

    MATH  Google Scholar 

  14. H. Zhang, W. Tong: The cancellable range of rings. Arch. Math. 85 (2005), 327–334.

    Article  MATH  MathSciNet  Google Scholar 

Download references

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  1. Department of Mathematics, Hunan Normal University, Changsha, 410006, P. R. China

    Huanyin Chen

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  1. Huanyin Chen
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Correspondence to Huanyin Chen.

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Chen, H. Strong separativity over exchange rings. Czech Math J 58, 417–428 (2008). https://doi.org/10.1007/s10587-008-0024-9

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  • DOI: https://doi.org/10.1007/s10587-008-0024-9

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