Generalizing unit-regular rings and special clean elements




Unit-regular rings, Clean rings, Special clean elements, Weakly clean rings, Weakly unit-regular rings


As a strengthening of the definition of weakly clean rings, given by Šter in J. Algebra (2014), and as a common generalization of the classical unit-regular rings, we define and investigate the class of socalled weakly unit-regular rings as those rings R for which, for every element a ? R, there exist a unit u and an idempotent e such that a ? u ? e ? (1 ? e)Ra with aR ? eR = {0}. Some more exotic relationships with the well-known classes of clean, nil-clean and (strongly) ?-regular rings are demonstrated as well. In particular, an elementwise extension of the so-called ”special clean elements” by Khurana et al. in J. Algebra & Appl. (2020) is also processed.


G. Azumaya, “Strongly ?-regular rings”, Journal of faculty of science, Hokkaido University. Series I. Mathematics, vol. 13, no. 1, pp. 34–39, 1954, doi: 10.14492/hokmj/1530842562

V. P. Camillo and D. Khurana, “A characterization of unit regular rings”, Communications in algebra, vol. 29, no. 5, pp. 2293–2295, Apr. 2001, doi: 10.1081/AGB-100002185

H. Chen, “On almost unit-regular rings”, Communications in algebra, vol. 40, no. 9, pp. 3494–3506, Sep. 2012, doi: 10.1080/00927872.2011.590953

A. Y. M. Chin and K. T. Qua, “A note on weakly clean rings”, Acta mathematica hungarica, vol. 132, no. 1-2, pp. 113–116, Apr. 2011, doi: 10.1007/s10474-011-0100-8

P. V. Danchev, “Generalizing nil clean rings”, Bulletin Belgian Mathematical Society -Simon Stevin, vol. 25, no. 1, pp. 13-28, 2018, doi: 10.36045/bbms/1523412048

P. V. Danchev, “Uniqueness in von Neumann regular unital rings”, Palestine journal mathematics, vol. 7, no. 1, pp. 60-63, 2018. [On line]. Available:

P. V. Danchev and J. Šter, “Generalizing ?-regular rings”, Taiwanese journal mathematics, vol. 19, no. 6, pp. 1577-1592, 2015, doi: 10.11650/tjm.19.2015.6236

G. Ehrlich, “Unit-regular rings”, Portugaliae. mathematica, vol. 27, no. 4, pp. 209-212, 1968. [On line]. Available:

K. R. Goodearl, Von Neumann regular rings, 2nd ed. Malabar, FL: Krieger, 1991.

J. Han and W. K. Nicholson, “Extensions of clean rings”, Communication in algebra, vol. 29, no. 6, pp. 2589-2595, 2001, doi: 10.1081/AGB-100002409

D. Handelman, “Perspectivity and cancellation in regular rings”, Journal algebra, vol. 48, no. 1, pp. 1-16, Sep. 1977, doi: 10.1016/0021-8693(77)90289-7

R. E. Hartwig and J. Luh, “A note on the group structure of unit regular ring elements”, Pacific journal mathematics, vol. 71, no. 2, pp. 449-461, 1977. [On line]. Available:

M. Henriksen, “On a class of regular rings that are elementary divisor rings”, Archiv der mathematik, vol. 24, pp. 133-141, Dec. 1973, doi: 10.1007/BF01228189

D. Khurana, T.-Y. Lam, P. P. Nielsen, and J. Šter, “Special clean elements in rings”, Journal of algebra and its applications, vol. 19, no. 11, Art. ID. 2050208, 2020, doi: 10.1142/S0219498820502084

T.-Y. Lam, A first course in noncommutative rings, 2nd ed. New York, NY: Springer, 2001, doi: 10.1007/978-1-4419-8616-0

T. Y. Lam, Exercises in classical ring theory, 2nd ed. New York, NY: Springer, 2003, doi: 10.1007/b97448

T. Y. Lam and W. Murray, “Unit regular elements in corner rings”, Bulletin of the Hong Kong Mathematical Society, vol. 1, pp. 61-65, 1997. [On line]. Available:

W. K. Nicholson, “Lifting idempotents and exchange rings”, Transactions of the American Mathematical Society, vol. 229, pp. 269-278, 1977, doi: 10.1090/S0002-9947-1977-0439876-2

W. K. Nicholson, “Strongly clean rings and Fitting’s lemma”, Communications in algebra, vol. 27, no. 8, pp. 3583-3592, 1999, doi: 10.1080/00927879908826649

P. P. Nielsen and J. Šter, “Connections between unit-regularity, regularity, cleanness, and strong cleanness of elements and rings”, Transactions of the American Mathematical Society, vol. 370, no. 3, pp. 1759-1782, 2018, doi: 10.1090/tran/7080

J. Šter, “Corner rings of a clean ring need not be clean”, Communications in algebra, vol. 40, no. 4, pp. 1595-1604, 2012, doi: 10.1080/00927872.2011.551901

J. Šter, “Weakly clean rings”, Journal algebra, vol. 401, pp. 1-12, Mar. 2014, doi: 10.1016/j.jalgebra.2013.10.034

J. Šter, “Examples of strongly clean rings”, Communications in algebra, vol. 47, no. 11, pp. 4684-4696, 2019, doi: 10.1080/00927872.2019.1588980

A. A. Tuganbaev, Rings close to regular. Dordrecht: Kluwer Academic Publishers, 2002, doi: 10.1007/978-94-015-9878-1

J. V. Neumann, “Examples of continuous geometries”, Proceedings of the National Academy of Sciences, vol. 22, no. 2, pp. 101-108, Feb. 1936, doi: 10.1073/pnas.22.2.101

J.-C. Wei, “Weakly-Abel rings and weakly exchange rings”, Acta mathematica hungarica vol. 137, pp. 254-262, 2012, doi: 10.1007/s10474-012-0253-0



How to Cite

P. V. Danchev, “Generalizing unit-regular rings and special clean elements”, Proyecciones (Antofagasta, On line), vol. 39, no. 5, pp. 1123-1135, Oct. 2020.