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A Theorem on Unit-Regular Rings

  • Tsiu-Kwen Lee (a1) and Yiqiang Zhou (a2)
Abstract

Let R be a unit-regular ring and let σ be an endomorphism of R such that σ(e) = e for all e 2 = eR and let n ≥ 0. It is proved that every element of R[x ; σ]/(xn +1) is equivalent to an element of the form e 0 + e 1 x + · · · + enxn , where the ei are orthogonal idempotents of R. As an application, it is proved that R[x ; σ]/(xn +1) is left morphic for each n ≥ 0.

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References
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[1] Chen, J. and Zhou, Y., Morphic rings as trivial extensions. Glasgow Math. J. 47(2005), no. 1, 139148. doi:10.1017/S0017089504002125
[2] Ehrlich, G., Units and one-sided units in regular rings. Trans. Amer. Math. Soc. 216(1976), 8190. doi:10.2307/1997686
[3] Goodearl, K. R., von Neumann Regular Rings. Second edition. Robert E. Krieger Publishing, Malabar, FL, 1991.
[4] Lee, T.-K. and Zhou, Y., Morphic rings and unit-regular rings. J. Pure Appl. Algebra 210(2007), no. 2, 501510. doi:10.1016/j.jpaa.2006.10.005
[5] Nicholson, W. K. and Campos, E. Sánchez, Rings with the dual of the isomorphism theorem. J. Algebra 271(2004), no. 1, 391406. doi:10.1016/j.jalgebra.2002.10.001
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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