We introduce the notions of left and right cleanness and nil cleanness in rings showing their close relationships with the classical concepts of cleanness and nil cleanness. Specifically, it is proved that strongly clean rings are both L-clean and R-clean as well as strongly nil clean rings are both L-nil clean and R-nil clean. These two assertions somewhat strengthen well-known results due to Nicholson (Comm. Algebra, 1999) and Diesl (J. Algebra, 2013). Moreover, it is shown that L-nil cleanness (respectively, R-nil cleanness) is preserved modulo nil Jacobson radical as well as that this is still true for L-cleanness (respectively, R-cleanness), provided the Jacobson radical is nil.

Peter Vassilev Danchev, Ph.D. (Math.), Prof., Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev st., bl. 8, Sofia, 1113, Bulgaria, e-mail: danchev@math.bas.bg; pvdanchev@yahoo.com

Danchev P.V. Left-Right Cleanness and Nil Cleanness in Unital Rings. The Bulletin of Irkutsk State University. Series Mathematics, 2019, vol. 27, pp. 28-35. https://doi.org/10.26516/1997-7670.2019.27.28

1. Breaz S., Calugareanu G., Danchev P., Micu T. Nil-clean matrix rings. Lin. Algebra & Appl., 2013, vol. 439, pp. 3115-3119. https://doi.org/10.1016/j.laa.2013.08.027
2. Camillo V.P., Khurana D. A characterization of unit regular rings. Commun. Algebra, 2001, vol. 29, pp. 2293-2295. https://doi.org/10.1081/AGB-100002185
3. Camillo V.P., Yu H.P. Exchange rings, units and idempotents. Commun. Algebra, 1994, vol. 22, pp. 4737-4749. https://doi.org/10.1080/00927879408825098
4. Danchev P.V., Lam T.Y. Rings with unipotent units. Publ. Math. Debrecen, 2016, vol. 88, pp. 449-466. https://doi.org/10.5486/PMD.2016.7405
5. Diesl A.J. Nil clean rings. J. Algebra, 2013, vol. 383, pp. 197-211. https://doi.org/10.1016/j.jalgebra.2013.02.020
6. Lam T.Y. A First Course in Noncommutative Rings. Second Edition, Graduate Texts in Math., 2001, vol. 131, Springer-Verlag, Berlin-Heidelberg-New York. https://doi.org/10.1007/978-1-4419-8616-0
7. Nicholson W. K. Strongly clean rings and Fitting’s lemma. Commun. Algebra, 1999, vol. 27, pp. 3583-3592. https://doi.org/10.1007/978-1-4419-8616-0
8. Kosan T., Wang Z., Zhou Y. Nil-clean and strongly nil-clean rings. J. Pure and Appl. Algebra, 2016, vol. 220, pp. 633-646.
9. Nielsen P.P., Ster J. Connections between unit-regularity, regularity, cleanness and strong cleanness of elements and rings. Trans. Amer. Math. Soc., 2018, vol. 370, pp. 1759-1782. https://doi.org/10.1090/tran/7080
10. Wang Z., Chen J. On two open problems about strongly clean rings. Bull. Austral. Math. Soc., 2004, vol. 70, pp. 279-282. https://doi.org/10.1017/S0004972700034493