Characterizations of m-weak group inverses

Abstract

To characterize m-weak group inverses, several algebraic methods are used, such as the use of idempotents, one-side principal ideals, and units. Consider an element a within a unitary ring that possesses Drazin invertibility and an involution. This paper begins by outlining the conditions necessary for the existence of the m-weak group inverse of a. Moreover, it explores the criteria under which a can be considered pseudo core invertible and weak group invertible. In the context of a weak proper *-ring, it is proved that a is weak group invertible if, and only if, a^D can serve as the weak group inverse of au, where u represents a specially invertible element closely associated with a^D. The paper also introduces a counterexample to illustrate that a^D cannot universally serve as the pseudo core inverse of another element. This distinction underscores the nuanced differences between pseudo core inverses and weak group inverses. Ultimately, the discussion expands to include the commuting properties of weak group inverses, extending these considerations to m-weak group inverses. Several new conditions on commuting properties of generalized inverses are given. These results show that pseudo core inverses, weak group inverses, and m-weak group inverses are not only closely linked but also have significant differences that set them apart.

Keywords

m-weak group inverse / weak group inverse / Drazin inverse / commuting property

References

[1] Drazin M P. Pseudo-inverses in associative rings and semigroups[J]. The American Mathematical Monthly, 1958, 65(7): 506-514. DOI: 10.2307/2308576.

[2] Baksalary O M, Trenkler G. Core inverse of matrices[J]. Linear and Multilinear Algebra, 2010, 58(6): 681-697. DOI:10.1080/03081080902778222.

[3] Gao Y F, Chen J L. Pseudo core inverses in rings with involution[J]. Communications in Algebra, 2018, 46(1): 38-50. DOI: 10.1080/00927872.2016.1260729.

[4] Prasad K M, Mohana K. Core-EP inverse[J]. Linear and Multilinear Algebra, 2014, 62: 792-802. DOI: 10.1080/03081087.2013.791690.

[5] Rakić D S, Dinčić N Ď, Djordjević D S. Group, Moore-Penrose, core and dual core inverse in rings with involution[J]. Linear Algebra and its Applications, 2014, 463: 115-133. DOI: 10.1016/j.laa.2014.09.003.

[6] Wang H X, Chen J L. Weak group inverse[J]. Open Mathematics, 2018, 16: 1218-1232. DOI: 10.1515/math-2018-0100.

[7] Zhou M M, Chen J L, Zhou Y K. Weak group inverses in proper *-rings[J]. Journal of Algebra and Its Applications, 2020, 19(12): 2050238. DOI: 10.1142/S0219498820502382.

[8] Zhou Y K, Chen J L, Zhou M M. m-weak group inverses in a ring with involution[J]. RACSAM, 2020, 115(1): 2. DOI: 10.1007/s13398-020-00932-1.

[9] Li W D, Chen J L, Zhou Y K. Characterizations and representations of weak core inverses and m-weak group inverses[J]. Turkish Journal of Mathematics, 2023, 47: 1453-1468. DOI:10.55730/1300-0098.3440.

[10] Zhou M M, Chen J L, Zhou Y K, et al. Weak group inverses and partial isometries in proper *-rings[J]. Linear and Multilinear Algebra, 2022, 70(19): 4528-4543. DOI: 10.1080/03081087.2021.1884639.

[11] Zhou Y K, Chen J L. Weak core inverses and pseudo core inverses in a ring with involution[J]. Linear and Multilinear Algebra, 2022, 70(21): 6876-6890. DOI: 10.1080/03081087.2021.1971151.

[12] Drazin M P. Commuting properties of generalized inverses[J]. Linear and Multilinear Algebra, 2013, 61(12): 1675-1681. DOI: 10.1080/03081087.2012.753593.

[13] Zhu H H, Chen J L. Representations of the Drazin inverse involving idempotents in a ring[J]. Journal of Southeast University(English Edition), 2015, 31(3): 427-430. DOI: 10.3969/j.issn.1003-7985.2015.03.023.

[14] Zou H L, Chen J L. Further results on pseudo Drazin inverse in Banach algebras[J]. Journal of Southeast University(Natural Science Edition), 2017, 47(3): 626-630. DOI:10.3969/j.issn.1001-0505.2017.03.034. (in Chinese)

[15] Zou H L, Zeng Y D, Chen J L. Generalized Drazin invertibility of the product of elements in Banach algebras[J]. Journal of Southeast University(Natural Science Edition), 2023, 53(1): 182-186. DOI:10.3969/j.jssn.1001-0505.2023.01.022. (in Chinese)

Funding

The National Natural Science Foundation of China(12171083); The National Natural Science Foundation of China(12071070); Qing Lan Project of Jiangsu Province and the Postgraduate Research and Practice Innovation Program of Jiangsu Province(KYCX22_0231)