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Transactions of the American Mathematical Society

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Half-orthogonal sets of idempotents


Authors: Victor P. Camillo and Pace P. Nielsen
Journal: Trans. Amer. Math. Soc. 368 (2016), 965-987
MSC (2010): Primary 16U99; Secondary 16D70, 16N20
DOI: https://doi.org/10.1090/tran/6350
Published electronically: May 6, 2015
MathSciNet review: 3430355
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Abstract: We improve several results in the literature focused on lifting idempotents, by either removing the lifting hypothesis or weakening other assumptions. We prove that countable sets of idempotents, which are orthogonal modulo an enabling ideal, lift to orthogonal idempotents. Left associates of liftable idempotents also lift modulo the Jacobson radical. Additionally, we exhibit situations when half-orthogonal sets of idempotents can be orthogonalized by multiplying by a unit. We finish by proving a number of results on directs sums of modules with the exchange property.


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  • [1] M. Alkan, W. K. Nicholson, and A. Ç. Özcan, Strong lifting splits, J. Pure Appl. Algebra 215 (2011), no. 8, 1879-1888. MR 2776430 (2012c:16012), https://doi.org/10.1016/j.jpaa.2010.11.001
  • [2] A. L. S. Corner, On the exchange property in additive categories, Unpublished Manuscript (1973), 60 pages.
  • [3] Peter Crawley and Bjarni Jónsson, Direct decompositions of algebraic systems, Bull. Amer. Math. Soc. 69 (1963), 541-547. MR 0156808 (28 #52)
  • [4] Dinesh Khurana and R. N. Gupta, Endomorphism rings of Harada modules, Vietnam J. Math. 28 (2000), no. 2, 173-175. MR 1810081
  • [5] T. Y. Lam, A First Course in Noncommutative Rings, second ed., Graduate Texts in Mathematics, vol. 131, Springer-Verlag, New York, 2001. MR 1838439 (2002c:16001)
  • [6] T. Y. Lam, Exercises in Classical Ring Theory, second ed., Problem Books in Mathematics, Springer-Verlag, New York, 2003. MR 2003255 (2004g:16001)
  • [7] Saad H. Mohamed and Bruno J. Müller, Continuous and discrete modules, London Mathematical Society Lecture Note Series, vol. 147, Cambridge University Press, Cambridge, 1990. MR 1084376 (92b:16009)
  • [8] Saad H. Mohamed and Bruno J. Müller, $ \aleph $-exchange rings, Abelian groups, module theory, and topology (Padua, 1997) Lecture Notes in Pure and Appl. Math., vol. 201, Dekker, New York, 1998, pp. 311-317. MR 1651176 (2000c:16006)
  • [9] W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977), 269-278. MR 0439876 (55 #12757)
  • [10] W. K. Nicholson and Y. Zhou, Strong lifting, J. Algebra 285 (2005), no. 2, 795-818. MR 2125465 (2006f:16006), https://doi.org/10.1016/j.jalgebra.2004.11.019
  • [11] Pace P. Nielsen, Countable exchange and full exchange rings, Comm. Algebra 35 (2007), no. 1, 3-23. MR 2287550 (2007k:16009), https://doi.org/10.1080/00927870600936815
  • [12] Pace P. Nielsen, Square-free modules with the exchange property, J. Algebra 323 (2010), no. 7, 1993-2001. MR 2594659 (2011b:16015), https://doi.org/10.1016/j.jalgebra.2009.12.035
  • [13] Josef Stock, On rings whose projective modules have the exchange property, J. Algebra 103 (1986), no. 2, 437-453. MR 864422 (88e:16038), https://doi.org/10.1016/0021-8693(86)90145-6
  • [14] R. B. Warfield Jr., A Krull-Schmidt theorem for infinite sums of modules, Proc. Amer. Math. Soc. 22 (1969), 460-465. MR 0242886 (39 #4213)
  • [15] Julius M. Zelmanowitz, Radical endomorphisms of decomposable modules, J. Algebra 279 (2004), no. 1, 135-146. MR 2078391 (2005e:16051), https://doi.org/10.1016/j.jalgebra.2004.04.004
  • [16] Birge Zimmermann-Huisgen and Wolfgang Zimmermann, Classes of modules with the exchange property, J. Algebra 88 (1984), no. 2, 416-434. MR 747525 (85i:16040), https://doi.org/10.1016/0021-8693(84)90075-9

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Additional Information

Victor P. Camillo
Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242
Email: camillo@math.uiowa.edu

Pace P. Nielsen
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
Email: pace@math.byu.edu

DOI: https://doi.org/10.1090/tran/6350
Keywords: Exchange property, half-orthogonal, idempotents, Jacobson radical, semi-$T$-nilpotent
Received by editor(s): August 28, 2013
Received by editor(s) in revised form: December 11, 2013
Published electronically: May 6, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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