Rings whose elements are sums or minus sums of three commuting idempotents

P. V. Danchev

doi:10.15330/ms.49.2.138-143

We completely determine up to isomorphism those rings whose elements x have the specific property that x or -x is a sum of three commuting idempotents. This statement strengthens well-known results in the subject due to Hirano-Tominaga (Bull. Austral. Math. Soc., 1988), Ying et al. (Can. Math. Bull., 2016), Tang et al. (Lin. & Multilin. Algebra, 2018), Danchev (Boll. Un. Mat. Ital., 2019) and (Bull. Iran. Math. Soc., 2019).

1. M.-S. Ahn, D.D. Anderson, Weakly clean rings and almost clean rings, Rocky Mountain J. Math., 36 (2006), 783-798.

2. P.V. Danchev, Rings whose elements are represented by at most three commuting idempotents, Gulf J. Math., 6 (2018), 1-6.

3. P.V. Danchev, Rings whose elements are sums of three or minus sums of two commuting idempotents, Alban. J. Math., 12 (2018), 3-7.

4. P.V. Danchev, Rings whose elements are sums of three or differences of two commuting idempotents, Bull. Iran. Math. Soc., 45 (2019).

5. P.V. Danchev, Rings whose elements are sums or minus sums of two commuting idempotents, Boll. Un. Mat. Ital., 12 (2019).

6. P.V. Danchev, Rings whose elements are sums of four commuting idempotents, to appear.

7. P.V. Danchev, W.Wm. McGovern, Commutative weakly nil clean unital rings, J. Algebra 425 (2015), 410-422.

8. P.V. Danchev, E. Nasibi, The idempotent sum number and n-thin unital rings, Ann. Univ. Sci. Budapest, (Sect. Math.) 59 (2016), 85-98.

9. Y. Hirano, H. Tominaga, Rings in which every element is the sum of two idempotents, Bull. Austral. Math. Soc., 37 (1988), 161-164.

10. T.Y. Lam, A First Course in Noncommutative Rings, Second Edition, Graduate Texts in Math., Vol. 131, Springer-Verlag, Berlin-Heidelberg-New York, 2001.

11. G. Tang, Y. Zhou, H. Su, Matrices over a commutative ring as sums of three idempotents or three involutions, Lin. and Multilin. Algebra (2018).

12. Z. Ying, T. Kosan, Y. Zhou, Rings in which every element is a sum of two tripotents, Can. Math. Bull., 59 (2016), №3, 661-672.

	Rings whose elements are sums or minus sums of three commuting idempotents
Author	P. V. Danchev danchev@math.bas.bg, pvdanchev@yahoo.com Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria
Abstract	We completely determine up to isomorphism those rings whose elements x have the specific property that x or -x is a sum of three commuting idempotents. This statement strengthens well-known results in the subject due to Hirano-Tominaga (Bull. Austral. Math. Soc., 1988), Ying et al. (Can. Math. Bull., 2016), Tang et al. (Lin. & Multilin. Algebra, 2018), Danchev (Boll. Un. Mat. Ital., 2019) and (Bull. Iran. Math. Soc., 2019).
Keywords	Boolean rings; finite fields; Jacobson radical; idempotents
DOI	doi:10.15330/ms.49.2.138-143
Reference	1. M.-S. Ahn, D.D. Anderson, Weakly clean rings and almost clean rings, Rocky Mountain J. Math., 36 (2006), 783-798. 2. P.V. Danchev, Rings whose elements are represented by at most three commuting idempotents, Gulf J. Math., 6 (2018), 1-6. 3. P.V. Danchev, Rings whose elements are sums of three or minus sums of two commuting idempotents, Alban. J. Math., 12 (2018), 3-7. 4. P.V. Danchev, Rings whose elements are sums of three or differences of two commuting idempotents, Bull. Iran. Math. Soc., 45 (2019). 5. P.V. Danchev, Rings whose elements are sums or minus sums of two commuting idempotents, Boll. Un. Mat. Ital., 12 (2019). 6. P.V. Danchev, Rings whose elements are sums of four commuting idempotents, to appear. 7. P.V. Danchev, W.Wm. McGovern, Commutative weakly nil clean unital rings, J. Algebra 425 (2015), 410-422. 8. P.V. Danchev, E. Nasibi, The idempotent sum number and n-thin unital rings, Ann. Univ. Sci. Budapest, (Sect. Math.) 59 (2016), 85-98. 9. Y. Hirano, H. Tominaga, Rings in which every element is the sum of two idempotents, Bull. Austral. Math. Soc., 37 (1988), 161-164. 10. T.Y. Lam, A First Course in Noncommutative Rings, Second Edition, Graduate Texts in Math., Vol. 131, Springer-Verlag, Berlin-Heidelberg-New York, 2001. 11. G. Tang, Y. Zhou, H. Su, Matrices over a commutative ring as sums of three idempotents or three involutions, Lin. and Multilin. Algebra (2018). 12. Z. Ying, T. Kosan, Y. Zhou, Rings in which every element is a sum of two tripotents, Can. Math. Bull., 59 (2016), №3, 661-672.
Pages	138-143
Volume	49
Issue	2
Year	2018
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html