Weak core inverses and pseudo core inverses in a ring with involution. (English) Zbl 1516.16033
Summary: In [Linear Multilinear Algebra 68, No. 1, 177–192 (2020; Zbl 1431.15006)], D. E. Ferreyra et al. defined the weak core inverse of a complex matrix. We generalize it to a unitary ring with involution and investigate the relationship between the weak core inverse and the \(m\)-weak group inverse. Furthermore, we prove that an element is pseudo core invertible if this element is both \(\{1,3\}\)-invertible and weak group invertible. Then, we show that each nilpotent element is Moore-Penrose invertible if and only if each pseudo core invertible element is weak core invertible. Finally, we consider the conditions under which the weak core inverse of an element coincides with another generalized inverse.
MSC:
| 16U90 | Generalized inverses (associative rings and algebras) |
| 16W10 | Rings with involution; Lie, Jordan and other nonassociative structures |
Keywords:
weak core inverse; weak group inverse; pseudo core inverse; \(m\)-weak group inverse; \(\{1,3\}\)-inverseCitations:
Zbl 1431.15006References:
| [1] | Ben-Israel, A.; Greville, TNE., Generalized inverses: theory and applications (2003), New York: Springer-Verlag, New York · Zbl 1026.15004 |
| [2] | Hartwig, RE., Block generalized inverses, Arch Rational Mech Anal, 61, 197-251 (1976) · Zbl 0335.15004 |
| [3] | Koliha, JJ; Djordjević, D.; Cvetković, D., Moore-Penrose inverse in rings with involution, Linear Algebra Appl, 426, 371-381 (2007) · Zbl 1130.46032 |
| [4] | Penrose, R., A generalized inverse for matrices, Proc Cambridge Philos Soc, 51, 406-413 (1955) · Zbl 0065.24603 |
| [5] | Drazin, MP., Pseudo-inverses in associative rings and semigroups, Amer Math Monthly, 65, 506-514 (1958) · Zbl 0083.02901 |
| [6] | Baksalary, OM; Trenkler, G., Core inverse of matrices, Linear Multilinear Algebra, 58, 6, 681-697 (2010) · Zbl 1202.15009 |
| [7] | Manjunatha Prasad, K.; Mohana, KS., Core-EP inverse, Linear Multilinear Algebra, 62, 6, 792-802 (2014) · Zbl 1306.15006 |
| [8] | Gao, YF; Chen, JL., Pseudo core inverses in rings with involution, Comm Algebra, 46, 1, 38-50 (2018) · Zbl 1392.15005 |
| [9] | Ferreyra, DE; Levis, FE; Thome, N., Revisiting the core-EP inverse and its extension to rectangular matrices, Quaest Math, 41, 2, 265-281 (2018) · Zbl 1390.15010 |
| [10] | Ferreyra, DE; Levis, FE; Thome, N., Characterizations of k-commutative equalities for some outer generalized inverses, Linear Multilinear Algebra, 68, 1, 177-192 (2018) · Zbl 1431.15006 |
| [11] | Ma, HF; Stanimirovic, PS., Characterizations, approximation and perturbations of the core-EP inverse, Appl Math Comput, 359, 404-417 (2019) · Zbl 1428.15004 |
| [12] | Rakić, DS; Dinčić, NČ.; Djordjević, DS., Group, Moore-Penrose, core and dual core inverse in rings with involution, Linear Algebra Appl, 463, 115-133 (2014) · Zbl 1297.15006 |
| [13] | Shi, GQ; Chen, JL; Li, TT, Jacobson’s lemma and Cline’s formula for generalized inverses in a ring with involution, Comm Algebra, 48, 9, 3948-3961 (2020) · Zbl 1465.16042 |
| [14] | Xu, SZ; Chen, JL; Zhang, XX., New characterizations for core inverses in rings with involution, Front Math China, 12, 1, 231-246 (2017) · Zbl 1379.16029 |
| [15] | Wang, HX; Chen, JL., Weak group inverse, Open Math, 16, 1218-1232 (2018) · Zbl 1408.15006 |
| [16] | Zhou, MM; Chen, JL; Zhou, YK., Weak group inverses in proper ∗-rings, J Algebra Appl, 19, 12 (2020) · Zbl 1464.16037 |
| [17] | Zhou, YK; Chen, JL; Zhou, MM., m-weak group inverses in a ring with involution, Rev R Acad Cienc Exactas Fís Nat Ser A Mat RACSAM, 115, 1 (2021) · Zbl 1455.16036 |
| [18] | Ferreyra, DE; Levis, FE; Priori, AN, The weak core inverse, Aequat Math, 95, 2, 351-373 (2021) · Zbl 1465.15008 |
| [19] | Malik, SB; Thome, N., On a new generalized inverse for matrices of an arbitrary index, Appl Math Comput, 226, 575-580 (2014) · Zbl 1354.15003 |
| [20] | Gao, YF; Chen, JL., ∗-DMP elements in ∗-semigroups and ∗-rings, Filomat, 32, 3073-3085 (2018) · Zbl 1513.16067 |
| [21] | Jain, SK; Manjunatha Prasad, K., Right-left symmetry of \(####\) in regular rings, J Pure Appl Algebra, 133, 141-142 (1998) · Zbl 0929.16010 |
| [22] | Zhou, MM; Chen, JL., Characterizations and maximal classes of elements related to pseudo core inverses, Rev R Acad Cienc Exactas Fís Nat Ser A Mat RACSAM, 114, 2 (2020) · Zbl 1437.16037 |
| [23] | Wang, HX., Core-EP decomposition and its applications, Linear Algebra Appl, 508, 289-300 (2016) · Zbl 1346.15003 |
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