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The (b; c)-inverse in semigroups and rings with involution |
Xiaofeng CHEN, Jianlong CHEN() |
School of Mathematics, Southeast University, Nanjing 210096, China |
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Abstract We first prove that if a is both left (b; c)-invertible and left (c; b)- invertible, then a is both (b; c)-invertible and (c; b)-invertible in a *-monoid, which generalizes the recent result about the inverse along an element by L. Wang and D. Mosić [Linear Multilinear Algebra, Doi.org/10.1080/03081087. 2019.1679073], under the conditions (ab)* = ab and (ac)* = ac: In addition, we consider that ba is (c; b)-invertible, and at the same time ca is (b; c)-invertible under the same conditions, which extend the related results about Moore- Penrose inverses studied by J. Chen, H. Zou, H. Zhu, and P. Patrício [Mediterr J. Math., 2017, 14: 208] to (b; c)-inverses. As applications, we obtain that under condition (a2)* = a2; a is an EP element if and only if a is one-sided core invertible, if and only if a is group invertible.
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Keywords
(b
c)-inverse, inverse along an element, core inverse, EP element
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Corresponding Author(s):
Jianlong CHEN
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Issue Date: 05 February 2021
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1 |
O M Baksalary, G Trenkler. Core inverse of matrices. Linear Multilinear Algebra, 2010, 58(6): 681–697
https://doi.org/10.1080/03081080902778222
|
2 |
J Chen, H Zou, H Zhu, P Patrício. The one-sided inverse along an element in semi- groups and rings. Mediterr J Math, 2017, 14: 208
https://doi.org/10.1007/s00009-017-1017-4
|
3 |
M P. DrazinA class of outer generalized inverses. Linear Algebra Appl, 2012, 436: 1909–1923
https://doi.org/10.1016/j.laa.2011.09.004
|
4 |
M P Drazin. Left and right generalized inverses. Linear Algebra Appl, 2016, 510: 64–78
https://doi.org/10.1016/j.laa.2016.08.010
|
5 |
R E. HartwigBlock generalized inverses. Arch Ration Mech Anal, 1976, 61: 197–251
https://doi.org/10.1007/BF00281485
|
6 |
Y Ke, Z Wang, J Chen. The (b; c)-inverse for products and lower triangular matrices. J Algebra Appl, 2017, 16(12): 1750222
https://doi.org/10.1142/S021949881750222X
|
7 |
J J Koliha, P. PatrícioElements of rings with equal spectral idempotents. J Aust Math Soc, 2002, 72: 137–152
https://doi.org/10.1017/S1446788700003657
|
8 |
T Li, J Chen. Characterizations of core and dual core inverses in rings with involution. Linear Multilinear Algebra, 2018, 66(4): 717–730
https://doi.org/10.1080/03081087.2017.1320963
|
9 |
X Mary. On generalized inverses and Green's relations. Linear Algebra Appl, 2011, 434: 1836–1844
https://doi.org/10.1016/j.laa.2010.11.045
|
10 |
D Mosić, D S. DjordievícFurther results on the reverse order law for the Moore-Penrose inverse in rings with involution. Appl Math Comput, 2011, 218: 1478–1483
https://doi.org/10.1016/j.amc.2011.06.040
|
11 |
D S Rakíc, N C Dinčíc, D S Djordievíc. Group, Moore-Penrose, core and dual core inverse in rings with involution. Linear Algebra Appl, 2014, 463: 115–133
https://doi.org/10.1016/j.laa.2014.09.003
|
12 |
L Wang, D Mosić. The one-sided inverse along two elements in rings. Linear Multi- linear Algebra, Doi.org/10.1080/03081087.2019.1679073
|
13 |
S, Xu J Chen, X Zhang. New characterizations for core and dual core inverses in rings with involution. Front Math China, 2017, 12(1): 231–246
https://doi.org/10.1007/s11464-016-0591-2
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14 |
H Zhu, J Chen, P. PatrícioFurther results on the inverse along an element in semi- groups and rings. Linear Multilinear Algebra, 2016, 64(3): 393{403
https://doi.org/10.1080/03081087.2015.1043716
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15 |
H Zou, J Chen, T Li, Y. GaoCharacterizations of core and dual core inverses in rings with involution. Bull Malays Math Sci Soc, 2018, 41(4): 1835–1857
https://doi.org/10.1007/s40840-016-0430-3
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