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The Nilpotent Regular Element Problem

  • Pere Ara (a1) and Kevin C. O'Meara (a2)

Abstract

We use George Bergman's recent normal form for universally adjoining an inner inverse to show that, for general rings, a nilpotent regular element $x$ need not be unit-regular. This contrasts sharply with the situation for nilpotent regular elements in exchange rings (a large class of rings), and for general rings when all powers of the nilpotent element $x$ are regular.

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[1] Ara, P., Strongly n-regular rings have stable range one. Proc. Amer. Math. Soc. 124(1996), 32933298. http://dx.doi.org/10.1090/S0002-9939-96-03473-9
[2] Ara, P., Goodearl, K. R., O'Meara, K. C., and Pardo, E., Separative cancellation for protective modules over exchange rings. Israel J. Math. 105(1998), 105137. http://dx.doi.org/10.1007/BF02780325
[3] Bergman, G. M., Adjoining a universal inner inverse to a ring element.J. Algebra 449(2016), 355399. http://dx.doi.org/=10.1016/j.jalgebra.2015.11.008
[4] Beidar, K. I., O'Meara, K. C., and Raphael, R. M., On uniform diagonalisation of matrices over regular rings and one-accessible regular algebras. Comm. Algebra 32(2004), 35433562. http://dx.doi.org/10.1081/ACB-120039630
[5] Goodearl, K. R., Von Neumann regular rings. Second edition. Krieger, Malabar, 1991.
[6] Lam, T. Y., Excursions in ring theory. Forthcoming.
[7] Nielsen, P. P. and Ster, J., Connections between unit-regularity, regularity, cleanness, and strong cleanness of elements and rings. arxiv:1510.03305v1.
[8] O'Meara, K. C., Clark, J., and Vinsonhaler, C. I., Advanced topics in linear algebra: weaving matrix problems through the Weyr form. Oxford University Press, Oxford, 2011.
[9] Yu, H.-P., On strongly pi-regular rings of stable range one. Bull. Austral.Math. Soc. 51(1995), 433437. http://dx.doi.org/10.1017/S0004972700014258
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The Nilpotent Regular Element Problem

  • Pere Ara (a1) and Kevin C. O'Meara (a2)

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