The $(j,m)$-core inverse in rings with involution

Year 2020, Volume: 49 Issue: 5, 1676 - 1685, 06.10.2020

Sanzhang Xu Jianlong Chen Dijana Mosić

https://doi.org/10.15672/hujms.701870

Abstract

Let $R$ be a unital ring with involution. The $(j,m)$-core inverse of a complex matrix was extended to an element in $R$. New necessary and sufficient conditions such that an element in $R$ to be $(j,m)$-core invertible are given. Moreover, several additive and product properties of two $(j,m)$-core invertible elements are investigated and a order related to the $(j,m)$-core inverse is introduced.

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Keywords

Moore-Penrose inverse , (j-m) core inverse , EP element

References

• [1] O.M. Baksalary and G. Trenkler. Core inverse of matrices, Linear Multilinear Algebra, 58(6), 681–697, 2010.
• [2] O.M. Baksalary and G. Trenkler, On a generalized core inverse, Appl. Math. Comput. 236, 450–457, 2014.
• [3] M.P. Drazin, Pseudo-inverses in associative rings and semigroup, Amer. Math. Monthly, 65, 506–514, 1958.
• [4] R.E. Hartwig, Block generalized inverses, Arch. Retional Mech. Anal. 61(3), 197–251, 1976.
• [5] S.B. Malik and N. Thome, On a new generalized inverse for matrices of an arbitrary index, Appl. Math. Comput. 226, 575–580, 2014.
• [6] D. Mosić and D.S. Djordjević, Moore-Penrose-invertible normal and Hermitian elements in rings, Linear Algebra Appl. 431, 732–745, 2009.
• [7] P. Patrício and R. Puystjens, Drazin-Moore-Penrose invertiblity in rings, Linear Algebra Appl. 389, 159–173, 2004.
• [8] D.S. Rakić, N.Č. Dinčić and D.S. Djordjević, Group, Moore-Penrose, core and dual core inverse in rings with involution, Linear Algebra Appl. 463, 115–133, 2014.
• [9] H. Schwerdtfeger, Introduction to Linear Algebra and the Theory of Matrices, P. Noordhoff, Groningen, 1950.
• [10] J. von Neumann, On regular rings, Proc. Nati. Acad. Sci. U.S.A. 22 (12), 707–713, 1936.
• [11] H.X. Wang and X.J. Liu, A partial order on the set of complex matrices with index one, Linear Multilinear Algebra, 66 (1), 206–216, 2018.
• [12] S.Z. Xu, J.L. Chen and J. Benítez, EP elements in rings with involution, Bull. Malays. Math. Sci. Soc. 42, 3409–3426, 2019.
• [13] S.Z. Xu, J.L. Chen, J. Benítez and D.G. Wang, Generalized core inverse of matrices, Miskolc Math. Notes, 20 (1), 565–584, 2019.
• [14] S.Z. Xu, J.L. Chen and X.X. Zhang, New characterizations for core and dual core inverses in rings with involution, Front. Math. China. 12 (1), 231–246, 2017.