EP Elements in Rings with Involution

[EN] Let R be a unital ring with involution. We first show that the EP elements in R can be characterized by three equations. Namely, let a. R, then a is EP if and only if there exists x. R such that (xa)* = xa, xa(2) = a and ax(2) = x. Any EP element in R is core invertible and Moore-Penrose invertible. We give more equivalent conditions for a core (Moore-Penrose) invertible element to be an EP element. Finally, any EP element is characterized in terms of the n-EP property, which is a generalization of the bi-EP property.This research is supported by the National Natural Science Foundation of China (No. 11771076). The first author is grateful to China Scholarship Council for giving him a purse for his further study in Universitat Politecnica de Valencia, Spain.Xu, S.; Chen, J.; Benítez López, J. (2019). EP Elements in Rings with Involution. Bulletin of the Malaysian Mathematical Sciences Society. 42(6):3409-3426. https://doi.org/10.1007/s40840-019-00731-xS34093426426Baksalary, O.M., Trenkler, G.: Core inverse of matrices. Linear Multilinear Algebra 58(6), 681–697 (2010)Benítez, J.: Moore–Penrose inverses and commuting elements of $C^{*}$-algebras. J. Math. Anal. Appl. 345(2), 766–770 (2008)Bhaskara Rao, K.P.S.: The Theory of Generalized Inverses Over Commutative Rings. Taylor and Francis, London (2002)Boasso, E.: On the Moore–Penrose inverse, EP Banach space operators, and EP Banach algebra elements. J. Math. Anal. Appl. 339(2), 1003–1014 (2008)Chen, W.X.: On EP elements, normal elements and partial isometries in rings with involution. Electron. J. Linear Algebra 23, 553–561 (2012)Drivaliaris, D., Karanasios, S., Pappas, D.: Factorizations of EP operators. Linear Algebra Appl. 429, 1555–1567 (2008)Hartwig, R.E.: Block generalized inverses. Arch. Retion. Mech. Anal. 61(3), 197–251 (1976)Hartwig, R.E., Spindelböck, K.: Matrices for which $A^*$ and $A^{\dagger }$ commute. Linear Multilinear Algebra 14(3), 241–256 (1983)Koliha, J.J., Patrício, P.: Elements of rings with equal spectral idempotents. J. Aust. Math. Soc. 72(1), 137–152 (2002)Mosić, D., Djordjević, D.S., Koliha, J.J.: EP elements in rings. Linear Algebra Appl. 431, 527–535 (2009)Mosić, D., Djordjević, D.S.: New characterizations of EP, generalized normal and generalized Hermitian elements in rings. Appl. Math. Comput. 218, 6702–6710 (2012)Patrício, P., Puystjens, R.: Drazin–Moore–Penrose invertiblity in rings. Linear Algebra Appl. 389, 159–173 (2004)Rakić, D.S., Dinčić, Nebojša Č., Djordjević, D.S.: Group, Moore–Penrose, core and dual core inverse in rings with involution. Linear Algebra Appl. 463, 115–133 (2014)von Neumann, J.: On regular rings. Proc. Natl. Acad. Sci. USA 22(12), 707–713 (1936

Generalized core inverses of matrices

[EN] n this paper, we introduce two new generalized inverses of matrices, namely, the -core inverse and the (j, m)-core inverse. The -core inverse of a complex matrix extends the notions of the core inverse defined by Baksalary and Trenkler [1] and the core-EP inverse defined by Manjunatha Prasad and Mohana [10]. The (j, m)-core inverse of a complex matrix extends the notions of the core inverse and the DMP-inverse defined by Malik and Thome [9]. Moreover, the formulae and properties of these two new concepts are investigated by using matrix decompositions and matrix powers.This research is supported by the National Natural Science Foundation of China (NO. 11771076 and No. 11471186). The first author is grateful to China Scholarship Council for giving him a purse for his further study in Universitat Politecnica de Valencia, Spain.Xu, S.; Chen, J.; Benítez López, J.; Wang, D. (2019). Generalized core inverses of matrices. Miskolc Mathematical Notes (Online). 20(1):565-584. https://doi.org/10.18514/MMN.2019.2594S56558420

Rank equalities related to a class of outer generalized inverse

[EN] In 2012, Drazin introduced a class of outer generalized inverse in a ring R, the (b, c)-inverse of a for a,b,c is an element of R and denoted by a(parallel to(b,c)). In this paper, rank equalities of A(k)A(parallel to(B,C)) - A parallel to((B,C))A(k) and (A*)(k)A(parallel to(B,C) )-( )A(parallel to(B,C))(A*)(k )are obtained. As applications, we investigate equivalent conditions for the equalities (A*)(k)A(parallel to(B,C)) = A(parallel to(B,C))(A*)(k) and A(k)A(parallel to(B,C)) = A(parallel to(B,C))A(k). As corollaries we obtain rank equalities related to the Moore-Penrose inverse, the core inverse, and the Drazin inverse. The paper finishes with some rank equalities involving different expressions containing A(parallel to(B,C)).The authors wish to thank the editor and reviewers sincerely for their constructive comments and suggestions that have improved the quality of the paper. The second author is grateful to China Scholarship Council for giving him a scholarship for his further study in Universitat Politècnica de València, Spain. This research is supported by the National Natural Science Foundation of China (No. 11771076), the Fundamental Research Funds for the Central Universities (no. KYCX 0055), the Postgraduate Research & Practice Innovation Program of Jiangsu Province (no. KYCX 0055). The second author is supported by the Natural Science Foundation of Jiangsu Education Committee (No. 19KJB110005) and the Natural Science Foundation of Jiangsu Province of China (No. BK20191047).Chen, J.; Xu, S.; Benítez López, J.; Chen, X. (2019). Rank equalities related to a class of outer generalized inverse. Filomat (Online). 33(17):5611-5622. https://doi.org/10.2298/FIL1917611CS56115622331

5,13-Disulfamoyl-1,9-diazatetracyclo[7.7.1.02,7.010,15]heptadeca-2(7),3,5,10,12,14-hexaen-1-ium chloride

In the title salt, C15H17N4O4S2 +·Cl−, the chloride anion is disordered over two positions with occupancies of 0.776 (6) and 0.224 (6). The cation adopts an L shape and the dihedral angle between the benzene rings is 82.5 (3)°. In the crystal, inversion dimers of cations linked by pairs of N—H⋯N hydrogen bonds occur, with the bond arising from the protonated N atom. The cationic dimers are linked into chains via the disordered chloride ions by way of N—H⋯Cl hydrogen bonds and N—H⋯O, C—H⋯O and C—H⋯Cl inter­actions also occur, which help to consolidate the three-dimensional network