Article
| Title: | Certain additive decompositions in a noncommutative ring (English) |
| Author: | Chen, Huanyin |
| Author: | Sheibani, Marjan |
| Author: | Bahmani, Rahman |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 72 |
| Issue: | 4 |
| Year: | 2022 |
| Pages: | 1217-1226 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | We determine when an element in a noncommutative ring is the sum of an idempotent and a radical element that commute. We prove that a $2\times 2$ matrix $A$ over a projective-free ring $R$ is strongly $J$-clean if and only if $A\in J (M_2(R))$, or $I_2-A\in J(M_2(R))$, or $A$ is similar to $\left (\smallmatrix 0&\lambda \\ 1&\mu \endsmallmatrix \right )$, where $\lambda \in J(R)$, $\mu \in 1+J(R)$, and the equation $x^2-x\mu -\lambda =0$ has a root in $J(R)$ and a root in $1+J(R)$. We further prove that $f(x)\in R[[x]]$ is strongly $J$-clean if $f(0)\in R$ be optimally $J$-clean. (English) |
| Keyword: | idempotent matrix |
| Keyword: | nilpotent matrix |
| Keyword: | projective-free ring |
| Keyword: | quadratic equation |
| Keyword: | power series |
| MSC: | 15A09 |
| MSC: | 16E50 |
| MSC: | 16U60 |
| idZBL: | Zbl 07655796 |
| idMR: | MR4517609 |
| DOI: | 10.21136/CMJ.2022.0039-22 |
| . | |
| Date available: | 2022-11-28T11:44:54Z |
| Last updated: | 2023-04-11 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151143 |
| . | |
| Reference: | [1] Anderson, D. D., Camillo, V. P.: Commutative rings whose elements are a sum of a unit and idempotent.Commun. Algebra 30 (2002), 3327-3336. Zbl 1083.13501, MR 1914999, 10.1081/AGB-120004490 |
| Reference: | [2] Ashrafi, N., Nasibi, E.: Strongly $J$-clean group rings.Proc. Rom. Acad., Ser. A, Math. Phys. Tech. Sci. Inf. Sci. 14 (2013), 9-12. Zbl 1313.16036, MR 3038863 |
| Reference: | [3] Chen, H.: Rings Related Stable Range Conditions.Series in Algebra 11. World Scientific, Hackensack (2011). Zbl 1245.16002, MR 2752904, 10.1142/8006 |
| Reference: | [4] Chen, H.: Strongly $J$-clean matrices over local rings.Commun. Algebra 40 (2012), 1352-1362. Zbl 1244.16024, MR 2912989, 10.1080/00927872.2010.551529 |
| Reference: | [5] Danchev, P. V., McGovern, W. W.: Commutative weakly nil clean unital rings.J. Algebra 425 (2015), 410-422. Zbl 1316.16028, MR 3295991, 10.1016/j.jalgebra.2014.12.003 |
| Reference: | [6] Diesl, A. J., Dorsey, T. J.: Strongly clean matrices over arbitrary rings.J. Algebra 399 (2014), 854-869. Zbl 1310.16023, MR 3144615, 10.1016/j.jalgebra.2013.08.044 |
| Reference: | [7] Dorsey, T. J.: Cleanness and Strong Cleanness of Rings of Matrices: Ph.D. Thesis.University of California, Berkeley (2006). MR 2709133 |
| Reference: | [8] Fan, L., Yang, X.: A note on strongly clean matrix rings.Commun. Algebra 38 (2010), 799-806. Zbl 1191.16029, MR 2650370, 10.1080/00927870802570693 |
| Reference: | [9] Koşan, M. T., Yildirim, T., Zhou, Y.: Rings whose elements are the sum of a tripotent and an element from the Jacobson radical.Can. Math. Bull. 62 (2019), 810-821. Zbl 07128566, MR 4028489, 10.4153/S0008439519000092 |
| Reference: | [10] Shifflet, D. R.: Optimally Clean Rings: Ph.D. Thesis.Bowling Green State University, Bowling Green (2011). MR 3121884 |
| Reference: | [12] Yang, X., Zhou, Y.: Strong cleanness of the $2\times 2$ matrix ring over a general local ring.J. Algebra 320 (2008), 2280-2290. Zbl 1162.16017, MR 2437500, 10.1016/j.jalgebra.2008.06.012 |
| Reference: | [13] Zhu, H., Zou, H., Patrício, P.: Generalized inverses and their relations with clean decompositions.J. Algebra Appl. 18 (2019), Article ID 1950133, 9 pages. Zbl 1453.16039, MR 3977794, 10.1142/S0219498819501330 |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
