The diamond partial order in rings

This is an author's accepted manuscript of an article published in " Linear and Multilinear Algebra"; Volume 62, Issue 3, 2014; copyright Taylor & Francis; available online at: http://dx.doi.org/10.1080/03081087.2013.779272In this paper we introduce a new partial order on a ring, namely the diamond partial order. This order is an extension of a partial order defined in a matrix setting in [J.K. Baksalary and J. Hauke, A further algebraic version of Cochran’s theorem and matrix partial orderings, Linear Algebra and its Applications, 127, 157–169, 1990]. We characterize the diamond partial order on rings and study its relationships with other partial orders known in the literature. We also analyse successors, predecessors and maximal elements under the diamond order.The first and third authors have been partially supported by Ministry of Education of Spain, grant DGI MTM2010-18228 and the third one by Universidad Nacional de La Pampa, Facultad de Ingenieria (grant Resol. No 049/11). The second author was financed by FEDER Funds through 'Programa Operacional Factores de Competitividade - COMPETE' and by Portuguese Funds through FCT - 'Fundacao para a Ciencia e a Tecnologia', within the project PEst-C/MAT/UI0013/2011.Lebtahi Ep-Kadi-Hahifi, L.; Patricio, P.; Thome, N. (2014). The diamond partial order in rings. Linear and Multilinear Algebra. 62(3):386-395. https://doi.org/10.1080/03081087.2013.779272386395623Mitra, S. K., & Bhimasankaram, P. (2010). MATRIX PARTIAL ORDERS, SHORTED OPERATORS AND APPLICATIONS. SERIES IN ALGEBRA. doi:10.1142/9789812838452Baksalary, J. K., & Hauke, J. (1990). A further algebraic version of Cochran’s theorem and matrix partial orderings. Linear Algebra and its Applications, 127, 157-169. doi:10.1016/0024-3795(90)90341-9Patrício P, Mendes Araujo C. Moore-Penrose invertibility in involutory rings: the caseaa†=bb†. Linear and Multilinear Algebra. 2010;58:445–452.Blackwood, B., Jain, S. K., Prasad, K. M., & Srivastava, A. K. (2009). Shorted Operators Relative to a Partial Order in a Regular Ring. Communications in Algebra, 37(11), 4141-4152. doi:10.1080/00927870902828629Baksalary, J. K., Baksalary, O. M., & Liu, X. (2003). Further properties of the star, left-star, right-star, and minus partial orderings. Linear Algebra and its Applications, 375, 83-94. doi:10.1016/s0024-3795(03)00609-8Baksalary, J. K., Baksalary, O. M., Liu, X., & Trenkler, G. (2008). Further results on generalized and hypergeneralized projectors. Linear Algebra and its Applications, 429(5-6), 1038-1050. doi:10.1016/j.laa.2007.03.029Hauke, J., Markiewicz, A., & Szulc, T. (2001). Inter- and extrapolatory properties of matrix partial orderings. Linear Algebra and its Applications, 332-334, 437-445. doi:10.1016/s0024-3795(01)00294-4Mosić, D., & Djordjević, D. S. (2012). Some results on the reverse order law in rings with involution. Aequationes mathematicae, 83(3), 271-282. doi:10.1007/s00010-012-0125-2Mosić, D., & Djordjević, D. S. (2011). Further results on the reverse order law for the Moore–Penrose inverse in rings with involution. Applied Mathematics and Computation, 218(4), 1478-1483. doi:10.1016/j.amc.2011.06.040Tošić, M., & Cvetković-Ilić, D. S. (2012). Invertibility of a linear combination of two matrices and partial orderings. Applied Mathematics and Computation, 218(9), 4651-4657. doi:10.1016/j.amc.2011.10.05