[1] Li Wende, Chen Jianlong,. Centrally clean elements and central Drazin inverses in a ring [J]. Journal of Southeast University (English Edition), 2022, 38 (3): 315-322. [doi:10.3969/j.issn.1003-7985.2022.03.014]
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Centrally clean elements and central Drazin inverses in a ring(
)
环中的中心clean元和中心Drazin逆
)Journal of Southeast University (English Edition)[ISSN:1003-7985/CN:32-1325/N]
- Volumn:
- 38
- Issue:
- 2022 3
- Page:
- 315-322
- Research Field:
- Mathematics, Physics, Mechanics
- Publishing date:
- 2022-09-20
Info
- Title:
- Centrally clean elements and central Drazin inverses in a ring
- 环中的中心clean元和中心Drazin逆
- Author(s):
- Li Wende; Chen Jianlong
- School of Mathematics, Southeast University, Nanjing 211189, China
- Keywords:
- centrally clean element; centrally clean ring; central Drazin inverse; central group inverse
- PACS:
- O153.3
- DOI:
- 10.3969/j.issn.1003-7985.2022.03.014
- Abstract:
- Element a in ring R is called centrally clean if it is the sum of central idempotent e and unit u. Moreover, a=e+u is called a centrally clean decomposition of a and R is called a centrally clean ring if every element of R is centrally clean. First, some characterizations of centrally clean elements are given. Furthermore, some properties of centrally clean rings, as well as the necessary and sufficient conditions for R to be a centrally clean ring are investigated. Centrally clean rings are closely related to the central Drazin inverses. Then, in terms of centrally clean decomposition, the necessary and sufficient conditions for the existence of central Drazin inverses are presented. Moreover, the central cleanness of special rings, such as corner rings, the ring of formal power series over ring R, and a direct product ∏Rα of ring Rα, is analyzed. Furthermore, the central group invertibility of combinations of two central idempotents in the algebra over a field is investigated. Finally, as an application, an example that lists all invertible, central group invertible, group invertible, central Drazin invertible elements, and centrally clean elements of the group ring Z2S3 is given.
- 如果环R中元素a是一个中心幂等元e和一个可逆元u的和, 那么称a是中心clean元, 并且称a=u+e为a的一个中心clean分解.如果环R中所有元素都是中心clean元, 则称环R是中心clean环.首先, 给出了中心clean元的等价刻画, 并进一步研究了中心clean环的一些性质以及环R是中心clean环的充分必要条件.中心clean环与中心Drazin逆有着紧密的联系.接着, 从中心clean分解的角度给出了中心Drazin逆存在的充分必要条件, 并研究了角环、幂级数环和环Rα的笛卡尔积∏Rα等特殊环的中心clean性.此外, 还研究了一般域上代数中的2个中心幂等元组合的中心群可逆性.最后, 作为一个应用, 分别计算出了群环Z2S3中所有的可逆元、中心群可逆元、群可逆元、中心Drazin可逆元以及中心clean元.
References:
[1] Nicholson W K. Lifting idempotents and exchange rings[J].Transactions of the American Mathematical Society, 1977, 229: 269-278. DOI:10.1090/s0002-9947-1977-0439876-2.
[2] Nicholson W K, Varadarajan K. Countable linear transformations are clean[J]. Proceedings of the American Mathematical Society, 1998, 126(1): 61-64. DOI:10.1090/s0002-9939-98-04397-4.
[3] Nicholson W K. Strongly clean rings and fitting’s lemma[J]. Communications in Algebra, 1999, 27(8): 3583-3592. DOI:10.1080/00927879908826649.
[4] Han J, Nicholson W K. Extensions of clean rings[J].Communications in Algebra, 2001, 29(6): 2589-2595. DOI:10.1081/agb-100002409.
[5] Hiremath V A, Hegde S. On strongly clean rings [J]. International Journal of Algebra, 2011, 5(1): 31-36.
[6] Camillo V P, Yu H P. Exchange rings, units and idempotents[J].Communications in Algebra, 1994, 22(12): 4737-4749. DOI:10.1080/00927879408825098.
[7] Chen J L, Cui J. Two questions of L.Va�9A; on *-clean rings[J]. Bulletin of the Australian Mathematical Society, 2013, 88(3): 499-505. DOI:10.1017/s0004972713000117.
[8] Cui J, Wang Z. A note on strongly*-clean rings[J]. Journal of the Korean Mathematical Society, 2015, 52(4): 839-851. DOI:10.4134/jkms.2015.52.4.839.
[9] Li C N, Zhou Y Q. On strongly*-clean rings[J]. Journal of Algebra and Its Applications, 2011, 10(6): 1363-1370. DOI:10.1142/s0219498811005221.
[10] Va�9A; L. *-clean rings; some clean and almost clean Baer *-rings and von Neumann algebras[J]. Journal of Algebra, 2010, 324(12): 3388-3400. DOI:10.1016/j.jalgebra.2010.10.011.
[11] Zhang H B, Camillo V. On clean rings[J].Communications in Algebra, 2016, 44(6): 2475-2481. DOI:10.1080/00927872.2015.1053899.
[12] Nicholson W K, Zhou Y Q. Rings in which elements are uniquely the sum of an idempotent and a unit[J]. Glasgow Mathematical Journal, 2004, 46(2): 227-236. DOI:10.1017/s0017089504001727.
[13] Drazin M P. Pseudo-inverses in associative rings and semigroups[J]. The American Mathematical Monthly, 1958, 65(7): 506-514. DOI:10.1080/00029890.1958.11991949.
[14] Zhu H H, Zou H L, Patrício P. Generalized inverses and their relations with clean decompositions[J]. Journal of Algebra and Its Applications, 2019, 18(7): 1950133. DOI:10.1142/s0219498819501330.
[15] Liu X J, Wu L L, Yu Y M. The group inverse of the combinations of two idempotent matrices[J]. Linear and Multilinear Algebra, 2011, 59(1): 101-115. DOI:10.1080/03081081003717986.
[16] Cao Q H, Xie T, Zuo K Z. Discussions on the group inverses of combinations of two idempotent matrices [J]. Journal of Wuhan University(Natural Science Edition), 2018, 64(3): 262-268.(in Chinese)
[17] Chen J L. Algebraic theory of generalized inverses: Group inverses and Drazin inverses[J]. Journal of Nanjing University Mathematical Biquarterly, 2021, 38(1): 1-113. DOI:10.3969/j.issn.0469-5097.2021.01.01.
[18] Chen J L, Gao Y F, Li L F. Drazin invertibility in a certain finite-dimensional algebra generated by two idempotents[J]. Numerical Functional Analysis and Optimization, 2020, 41(14): 1804-1817. DOI:10.1080/01630563.2020.1813167.
[19] Chen J L, Zhuang G F, Wei Y M. The Drazin inverse of a sum of morphisms[J]. Acta Mathematica Scientia: Series A, 2009, 29(3): 538-552.(in Chinese)
[20] Cvetkovi’/c-Ili’/c D S, Wei Y M. Algebraic properties of generalized inverses [M]. Singapore: Springer, 2017: 89-93.
[21] Guo L, Chen J L, Zou H L. Representations for the Drazin inverse of the sum of two matrices and its applications[J]. Bulletin of the Iranian Mathematical Society, 2019, 45(3): 683-699. DOI:10.1007/s41980-018-0159-x.
[22] Xie T, Zuo K Z, Zheng L Z. Group inverse of combinations of two idempotent matrices[J]. Journal of Jilin University(Science Edition), 2016, 54(1): 45-53.(in Chinese)
[23] Zhang D C, Mosi’/c D, Guo L. The Drazin inverse of the sum of four matrices and its applications[J]. Linear and Multilinear Algebra, 2020, 68(1): 133-151. DOI:10.1080/03081087.2018.1500518.
[24] Zou H L, Mosi’/c D, Chen J L. The existence and representation of the Drazin inverse of a 2 × 2 block matrix over a ring[J]. Journal of Algebra and Its Applications, 2019, 18(11): 1950212. DOI:10.1142/s0219498819502128.
[25] Zhu H H, Chen J L. Additive and product properties of Drazin inverses of elements in a ring[J]. Bulletin of the Malaysian Mathematical Sciences Society, 2017, 40(1): 259-278. DOI:10.1007/s40840-016-0318-2.
[26] Zhou M M, Chen J L, Zhu X. The group inverse and core inverse of sums of two elements in a ring[J]. Communications in Algebra, 2020, 48(2): 676-690. DOI:10.1080/00927872.2019.1654497.
[27] Drazin M P. Commuting properties of generalized inverses[J]. Linear and Multilinear Algebra, 2013, 61(12): 1675-1681. DOI:10.1080/03081087.2012.753593.
[28] Wu C, Zhao L. Central Drazin inverses[J]. Journal of Algebra and Its Applications, 2019, 18(4): 1950065. DOI:10.1142/s0219498819500658.
[29] Zhao L, Wu C, Wang Y. One-sided central Drazin inverses[J]. Linear and Multilinear Algebra, 2022, 70(7): 1193-1206. DOI:10.1080/03081087.2020.1757601.
[30] Bhaskara Rao K P S. The theory of generalized inverses over commutative rings [M]. London: Taylor and Francis, 2002: 144-146.
Memo
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Biographies: Li Wende(1993—), male, Ph.D. candidate; Chen Jianlong(corresponding author), male, doctor, professor, jlchen@seu.edu.cn.
Foundation items: The National Natural Science Foundation of China(No. 12171083, 11871145, 12071070), the Qing Lan Project of Jiangsu Province.
Citation: Li Wende, Chen Jianlong. Centrally clean elements and central Drazin inverses in a ring[J].Journal of Southeast University(English Edition), 2022, 38(3):315-322.DOI:10.3969/j.issn.1003-7985.2022.03.014.