On Generalized Inverse And Partial Order In Rings

Moore-Penrose inverse (abbr. MP inverse) and Drazin inverse are the two most basic concepts in the theory of generalized inverse. People have carried out extensive research on the theory as well as on the applications of generalized inverses. In the process, many new notions such as group inverse, core inverse, DMP inverse, etc, and some new methods were introduced. Among other things, it is fruitful to relate the generalized inverses to partial order. These are established mainly for complex matrices, C*-algebra or algebra of operators. Recently, some scholars tried to generalize these from some specific algebras to a ring. In this regard, there are many problems needed further investigation. Note that some known techniques and methods based on matrix factorization, rank or norm are not available in the general setting of rings. In this article, we adopt some ring theoretical methods to study generalized inverses and partial orders in a ring R with a involution* in three aspects as follows.Firstly, we consider the generalized involutive elements and their linear combinations in the ring R. The notion of k-generalized involutive elements is introduced to extend that of generalized involutive matrices and hypergeneralized projectors. An element a in R is called k-generalized involutive if ak=at. A k-generalized involutive element in the ring R is characterized in terms of MP inverse and group inverse. Then, as applications to a complex algebra we obtain sufficient and necessary conditions under which a linear combination of two generalized involutive elements is generalized involutive. Some related results on complex matrices due to X.J. Liu, etc, are extended.Secondly, we investigate core inverse, core partial order and DMP inverse in the ring R. Some known results on complex matrices, including characterizations and properties of core inverse and core partial order, are generalized to the case for elements in the ring R. Then, we characterize and obtain some properties of the DMP inverse of elements in R. The relation between DMP inverse and (b, c)-inverse are also discussed. The results that we get here extend those relative work by S.B. Malik and some other authors.Finally, partial order and invertibility of a linear combination in an algebra A over an arbitrary field F are considered. Under some assumptions of *-orthogonality or some partial order on a, b ∈ A, the invertibility of the combination C1a+c2b or c1an+c26n are characterized. Meanwhile, the expressions of their inverses are also presented in a concise form. Thus, some related results on complex matrices due to M. Tosic, etc, are generalized.