Geodesic
Left-right cleanness and nil cleanness in unital rings
The Bulletin of Irkutsk State University. Series Mathematics, Tome 27 (2019), pp. 28-35
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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.
Keywords: clean rings, nil clean rings, L-clean rings, R-clean rings, L-nil clean rings, R-nil clean rings. 
@article{IIGUM_2019_27_a2,
     author = {P. V. Danchev},
     title = {Left-right cleanness and nil cleanness in unital rings},
     journal = {The Bulletin of Irkutsk State University. Series Mathematics},
     pages = {28--35},
     year = {2019},
     volume = {27},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IIGUM_2019_27_a2/}
}
TY  - JOUR
AU  - P. V. Danchev
TI  - Left-right cleanness and nil cleanness in unital rings
JO  - The Bulletin of Irkutsk State University. Series Mathematics
PY  - 2019
SP  - 28
EP  - 35
VL  - 27
UR  - http://geodesic.mathdoc.fr/item/IIGUM_2019_27_a2/
LA  - en
ID  - IIGUM_2019_27_a2
ER  - 
%0 Journal Article
%A P. V. Danchev
%T Left-right cleanness and nil cleanness in unital rings
%J The Bulletin of Irkutsk State University. Series Mathematics
%D 2019
%P 28-35
%V 27
%U http://geodesic.mathdoc.fr/item/IIGUM_2019_27_a2/
%G en
%F IIGUM_2019_27_a2
P. V. Danchev. Left-right cleanness and nil cleanness in unital rings. The Bulletin of Irkutsk State University. Series Mathematics, Tome 27 (2019), pp. 28-35. http://geodesic.mathdoc.fr/item/IIGUM_2019_27_a2/

[1] Breaz S., Cǎlugǎreanu G., Danchev P., Micu T., “Nil-clean matrix rings”, Lin. Algebra Appl., 439 (2013), 3115–3119 | DOI | MR | Zbl

[2] Camillo V. P., Khurana D., “A characterization of unit regular rings”, Commun. Algebra, 29 (2001), 2293–2295 | DOI | MR | Zbl

[3] Camillo V. P., Yu H. P., “Exchange rings, units and idempotents”, Commun. Algebra, 22 (1994), 4737–4749 | DOI | MR | Zbl

[4] Danchev P. V., Lam T. Y., “Rings with unipotent units”, Publ. Math. Debrecen, 88 (2016), 449–466 | DOI | MR | Zbl

[5] Diesl A. J., “Nil clean rings”, J. Algebra, 383 (2013), 197–211 | DOI | MR | Zbl

[6] Lam T. Y., A First Course in Noncommutative Rings, Graduate Texts in Math., 131, Second Edition, Springer-Verlag, Berlin–Heidelberg–New York, 2001 | DOI | MR | Zbl

[7] Nicholson W. K., “Strongly clean rings and Fitting's lemma”, Commun. Algebra, 27 (1999), 3583–3592 | DOI | MR | Zbl

[8] Koşan T., Wang Z., Zhou Y., “Nil-clean and strongly nil-clean rings”, J. Pure and Appl. Algebra, 220 (2016), 633–646 | DOI | MR | Zbl

[9] Nielsen P. P., Šter J., “Connections between unit-regularity, regularity, cleanness and strong cleanness of elements and rings”, Trans. Amer. Math. Soc., 370 (2018), 1759–1782 | DOI | MR | Zbl

[10] Wang Z., Chen J., “On two open problems about strongly clean rings”, Bull. Austral. Math. Soc., 70 (2004), 279–282 | DOI | MR | Zbl