# Institute of Mathematics

## Strongly 2-nil-clean rings with involutions

#### Huanyin Chen, Marjan Sheibani Abdolyousefi

###### Received June 15, 2017.   Published online August 6, 2018.

Abstract:  A $*$-ring $R$ is strongly 2-nil-$*$-clean if every element in $R$ is the sum of two projections and a nilpotent that commute. Fundamental properties of such $*$-rings are obtained. We prove that a $*$-ring $R$ is strongly 2-nil-$*$-clean if and only if for all $a\in R$, $a^2\in R$ is strongly nil-$*$-clean, if and only if for any $a\in R$ there exists a $*$-tripotent $e\in R$ such that $a-e\in R$ is nilpotent and $ea=ae$, if and only if $R$ is a strongly $*$-clean SN ring, if and only if $R$ is abelian, $J(R)$ is nil and $R/J(R)$ is $*$-tripotent. Furthermore, we explore the structure of such rings and prove that a $*$-ring $R$ is strongly 2-nil-$*$-clean if and only if $R$ is abelian and $R\cong R_1, R_2$ or $R_1\times R_2$, where $R_1/J(R_1)$ is a $*$-Boolean ring and $J(R_1)$ is nil, $R_2/J(R_2)$ is a $*$-Yaqub ring and $J(R_2)$ is nil. The uniqueness of projections of such rings are thereby investigated.
Keywords:  nilpotent; projection; $*$-tripotent ring; symmetry; strongly $*$-clean ring
Classification MSC:  16U99, 16E50, 16W10
DOI:  10.21136/CMJ.2018.0291-17

PDF available at:  Springer   Institute of Mathematics CAS

References:
[1] S. K. Berberian: Baer $*$-Rings. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 195, Springer, New York (1972). DOI 10.1007/978-3-642-15071-5 | MR 0429975 | Zbl 0242.16008
[2] H. Chen: Rings Related to Stable Range Conditions. Series in Algebra 11, World Scientific, Hackensack (2011). DOI 10.1142/9789814329729 | MR 2752904 | Zbl 1245.16002
[3] H. Chen, A. Harmanci, A. Ç. Özcan: Strongly $J$-clean rings with involutions. Ring Theory and Its Applications (D. V. Huynh, et al., eds.). Contemporary Mathematics 609, American Mathematical Society, Providence (2014), 33-44. DOI 10.1090/conm/609/12122 | MR 3204350 | Zbl 1296.16045
[4] H. Chen, M. Sheibani: Strongly 2-nil-clean rings. J. Algebra Appl. 16 (2017), Article ID 1750178, 12 pages. DOI 10.1142/S021949881750178X | MR 3661645 | Zbl 1382.16035
[5] J. Cui, Z. Wang: A note on strongly $*$-clean rings. J. Korean Math. Soc. 52 (2015), 839-851. DOI 10.4134/JKMS.2015.52.4.839 | MR 3369115 | Zbl 1327.16030
[6] P. V. Danchev: Weakly UU rings. Tsukuba J. Math. 40 (2016), 101-118. DOI 10.21099/tkbjm/1474747489 | MR 3550934 | Zbl 1377.16031
[7] P. V. Danchev: Invo-clean unital rings. Commun. Korean Math. Soc. 32 (2017), 19-27. DOI 10.4134/CKMS.c160054 | MR 3608475 | Zbl 1357.16054
[8] Y. Gao, J. Chen, Y. Li: Some $*$-clean group rings. Algebra Colloq. 22 (2015), 169-180. DOI 10.1142/S1005386715000152 | MR 3296765 | Zbl 1316.16018
[9] D. Han, Y. Ren, H. Zhang: On $*$-clean group rings over abelian groups. J. Algebra Appl. 16 (2017), Article ID 1750152, 11 pages. DOI 10.1142/S0219498817501523 | MR 3661619 | Zbl 1382.16018
[10] Y. Hirano, H. Tominaga: Rings in which every element is the sum of two idempotents. Bull. Aust. Math. Soc. 37 (1988), 161-164. DOI 10.1017/S000497270002668X | MR 0930784 | Zbl 0688.16015
[11] H. Huang, Y. Li, P. Yuan: On $*$-clean group rings II. Commun. Algebra 44 (2016), 3171-3181. DOI 10.1080/00927872.2015.1044106 | MR 3507177 | Zbl 1355.16019
[12] T. Koşan, Z. Wang, Y. Zhou: Nil-clean and strongly nil-clean rings. J. Pure Appl. Algebra 220 (2016), 633-646. DOI 10.1016/j.jpaa.2015.07.009 | MR 3399382 | Zbl 1335.16026
[13] Y. Li, M. M. Parmenter, P. Yuan: On $*$-clean group rings. J. Algebra Appl. 14 (2015), Article ID 1550004, 11 pages. DOI 10.1142/S0219498815500048 | MR 3257826 | Zbl 1318.16024
[14] C. Li, Y. Zhou: On strongly $*$-clean rings. J. Algebra Appl. 10 (2011), 1363-1370. DOI 10.1142/S0219498811005221 | MR 2864582 | Zbl 1248.16030
[15] Z. Ying, T. Koşan, Y. Zhou: Rings in which every element is a sum of two tripotents. Can. Math. Bull. 59 (2016), 661-672. DOI 10.4153/CMB-2016-009-0 | MR 3563747 | Zbl 1373.16067

Affiliations:   Huanyin Chen, Department of Mathematics, Hangzhou Normal University, No.2318,Yuhangtang Rd, Cangqian, Yuhang District, 311121, Hangzhou, Zhejiang Province, P. R. China, e-mail: huanyinchen@aliyun.com; Marjan Sheibani Abdolyousefi, (corresponding author), Women's University of Semnan (Farzanegan), Semnan, Iran, e-mail: sheibani@fgusem.ac.ir

PDF available at: