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. doi:10.1080/03081087.2013.779272S38639562