## Exchange elements in rings, and the equation $XA-BX=I$

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- by Dinesh Khurana, T. Y. Lam and Pace P. Nielsen PDF
- Trans. Amer. Math. Soc.
**369**(2017), 495-516

## Abstract:

The equation $XA-BX=I$ has been well studied in ring theory, operator theory, linear algebra, and other branches of mathematics. In this paper, we show that, in the case where $B^2=B$, the study of $XA-BX=I$ in a noncommutative ring $R$ leads to several new ways to view and to work with the exchange (or “suitable”) elements in $R$ in the sense of Nicholson. For any exchange element $A\in R$, we show that the set of idempotents $E\in R$ such that $E\in R A$ and $I-E\in R (I-A)$ is naturally parametrized by the roots of a certain left-right symmetric “exchange polynomial” associated with $A$. From the new viewpoints on exchange elements developed in this paper, the classes of clean and strongly clean elements in rings can also be better understood.## References

- P. Ara, K. R. Goodearl, K. C. O’Meara, and E. Pardo,
*Separative cancellation for projective modules over exchange rings*, Israel J. Math.**105**(1998), 105–137. MR**1639739**, DOI 10.1007/BF02780325 - Weixing Chen,
*A question on strongly clean rings*, Comm. Algebra**34**(2006), no. 7, 2347–2350. MR**2240371**, DOI 10.1080/00927870600550202 - P. M. Cohn,
*Free ideal rings and localization in general rings*, New Mathematical Monographs, vol. 3, Cambridge University Press, Cambridge, 2006. MR**2246388**, DOI 10.1017/CBO9780511542794 - A. L. S. Corner,
*On the exchange property in additive categories*, unpublished manuscript, Worcester College, Oxford, 1973. - Alexander J. Diesl,
*Nil clean rings*, J. Algebra**383**(2013), 197–211. MR**3037975**, DOI 10.1016/j.jalgebra.2013.02.020 - K. R. Goodearl and R. B. Warfield Jr.,
*Algebras over zero-dimensional rings*, Math. Ann.**223**(1976), no. 2, 157–168. MR**412230**, DOI 10.1007/BF01360879 - Juncheol Han and W. K. Nicholson,
*Extensions of clean rings*, Comm. Algebra**29**(2001), no. 6, 2589–2595. MR**1845131**, DOI 10.1081/AGB-100002409 - Nathan Jacobson,
*Structure of rings*, American Mathematical Society Colloquium Publications, Vol. 37, American Mathematical Society, 190 Hope Street, Providence, R.I., 1956. MR**0081264** - Dinesh Khurana and T. Y. Lam,
*Clean matrices and unit-regular matrices*, J. Algebra**280**(2004), no. 2, 683–698. MR**2090058**, DOI 10.1016/j.jalgebra.2004.04.019 - Dinesh Khurana, T. Y. Lam, and Pace P. Nielsen,
*Two-sided properties of elements in exchange rings*, Algebr. Represent. Theory**18**(2015), no. 4, 931–940. MR**3372125**, DOI 10.1007/s10468-015-9524-0 - T. Y. Lam,
*A first course in noncommutative rings*, 2nd ed., Graduate Texts in Mathematics, vol. 131, Springer-Verlag, New York, 2001. MR**1838439**, DOI 10.1007/978-1-4419-8616-0 - T. Y. Lam and Will Murray,
*Unit regular elements in corner rings*, Bull. Hong Kong Math. Soc.**1**(1997), no. 1, 61–65. MR**1466836** - Tsiu-Kwen Lee and Yiqiang Zhou,
*A class of exchange rings*, Glasg. Math. J.**50**(2008), no. 3, 509–522. MR**2451746**, DOI 10.1017/S0017089508004370 - Tsiu-Kwen Lee and Yiqiang Zhou,
*Clean index of rings*, Comm. Algebra**40**(2012), no. 3, 807–822. MR**2899909**, DOI 10.1080/00927872.2010.538781 - Saad H. Mohamed,
*Report on exchange rings*, Advances in ring theory, Trends Math., Birkhäuser/Springer Basel AG, Basel, 2010, pp. 239–255. MR**2664675**, DOI 10.1007/978-3-0346-0286-0_{1}6 - G. S. Monk,
*A characterization of exchange rings*, Proc. Amer. Math. Soc.**35**(1972), 349–353. MR**302695**, DOI 10.1090/S0002-9939-1972-0302695-2 - W. K. Nicholson,
*Lifting idempotents and exchange rings*, Trans. Amer. Math. Soc.**229**(1977), 269–278. MR**439876**, DOI 10.1090/S0002-9947-1977-0439876-2 - W. K. Nicholson,
*On exchange rings*, Comm. Algebra**25**(1997), no. 6, 1917–1918. MR**1446139**, DOI 10.1080/00927879708825962 - W. K. Nicholson,
*Strongly clean rings and Fitting’s lemma*, Comm. Algebra**27**(1999), no. 8, 3583–3592. MR**1699586**, DOI 10.1080/00927879908826649 - W. K. Nicholson and Y. Zhou,
*Strong lifting*, J. Algebra**285**(2005), no. 2, 795–818. MR**2125465**, DOI 10.1016/j.jalgebra.2004.11.019 - Janez Šter,
*Corner rings of a clean ring need not be clean*, Comm. Algebra**40**(2012), no. 5, 1595–1604. MR**2924469**, DOI 10.1080/00927872.2011.551901 - R. B. Warfield Jr.,
*A Krull-Schmidt theorem for infinite sums of modules*, Proc. Amer. Math. Soc.**22**(1969), 460–465. MR**242886**, DOI 10.1090/S0002-9939-1969-0242886-2 - R. B. Warfield Jr.,
*Exchange rings and decompositions of modules*, Math. Ann.**199**(1972), 31–36. MR**332893**, DOI 10.1007/BF01419573

## Additional Information

**Dinesh Khurana**- Affiliation: Department of Mathematics, Panjab University, Chandigarh 160 014, India
- MR Author ID: 658568
- Email: dkhurana@pu.ac.in
**T. Y. Lam**- Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
- MR Author ID: 109495
- Email: lam@math.berkeley.edu
**Pace P. Nielsen**- Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
- MR Author ID: 709329
- Email: pace@math.byu.edu
- Received by editor(s): October 22, 2014
- Received by editor(s) in revised form: January 8, 2015
- Published electronically: March 2, 2016
- © Copyright 2016 by the authors
- Journal: Trans. Amer. Math. Soc.
**369**(2017), 495-516 - MSC (2010): Primary 16E50, 16U99
- DOI: https://doi.org/10.1090/tran6652
- MathSciNet review: 3557782