Generalized Inverses And Related Partial Orders Over Rings

The theory of generalized inverse of matrices is an important subject of the study of matrices.The first concept of generalized inverse was introduced by E.H.Moore in 1920 and a simpler characterization of this generalized inverse was given in terms of four matrix equations by R.Penrose in 1955.Three years later,Drazin introduced another kind of generalized inverse in semigroups and rings,which is called Drazin inverse nowdays.The Moore-Penrose inverse and the Drazin inverse are two classic generalized inverses.With the development of science and technology,the theory of generalized inverses of matrices has got deep development and extensive applications.In recent years,some new concepts of generalized inverses were introduced,for example,the core inverse and(b,c)-inverse.The core inverse,Moore-Penrose inverse,(b,c)-inverse and EP element are hot topics in generalized inverses research field.Due to the applications in statistics,partial order theory based on generalized inverses also is a hot topic in generalized inverses research field.For example,the star partial order,the minus partial order,the sharp partial order and the core partial order have been introduced.In recent years,more and more scholars study the generalized inverses and partial orders based on generalized inverses over a ring.In this thesis,we mainly adopt some ring theoretic methods to study the theory of generalized inverses in rings.The main contents are arranged as follows:In chapter 2,existence criteria and expressions for the Moore-Penrose inverse are characterized by invertible,Hermite,projection elements and direct sums.we show that an element a in a ring R with involution is Moore-Penrose invertible if and only if a is well-supported.Therefore,the star-cancelable condition of the work of J.J.Koliha et al.can be dropped.Also,we show that the existence of the Moore-Penrose inverse can be characterized by some decompositions of the ring.In addition,the formulae of the Moore-Penrose inverse of an element in R are presented.Furthermore,some properties of projections related the Moore-Penrose inverse are investigated.In chapter 3,we investigate the core inverse over a ring with an involution.Firstly,we study the existence of the core inverses in terms of equations.We prove that the existence criterion of the core inverse can be characterized by three equations,which improved the work of D.S.Rakic et al.in Linear Algebra Appl.We answer when a group invertible element is core invertible.We also use invertible elements to characterize the existence of the core inverses.Secondly,several necessary and sufficient conditions which guarantee the absorption law and the reverse order law for core invertible elements hold are given.We also consider the core invertibility of triangular matrices over a ring with an involution.Finally,we investigate the additive property of two core invertible elements.Moreover,the formulae of the sum of two core invertible elements are presented.In chapter 4,existence criteria for the(b,c)-inverse are obtained and centralizer’s applications of the(b,c)-inverse are considered.We present explicit expressions for the(b,c)-inverse in terms of inner inverses.Let a,b,c ∈ R,it is well-known that the(b,c)-inverse of a is an outer {2}-inverse of a,but not an inner inverse of a.We answer the question when the(b,c)-inverse of a ∈R is an inner inverse of a in general.Let a,b,c,y∈R.We show that,y is an inner(b,c)-inverse of a if and only if a is regular,R = a° ⊕ bR and R = °a ⊕Rc.Moreover,rank equalities of the commutator AA‖(B,C)-A‖(B,C)A are obtained,where,B,C∈Cn×n.If A is(B,C)-invertible with(B,C)-inverse A‖(B,C),then we have rank(AA‖(B.C)-A‖(B,C)A)= rank,([CCA])+rank {[AB | B])-2 rank(B),which is an improvement of the works of Yonghui Liu et al.In chapter 5,we characterize the EP elements in a ring R by three equations.Namely,if a ∈R,then a is EP if and only if there exists x ∈ R such that(xa)*= xa,xa2 = a and ax2 = x.Many equivalent conditions for a core(Moore-Penrose)invertible element to be an EP element are given.We show that the condition aR = a*R in the work of P.Patricio and R.Puystjens on Linear Algebra Appl.can be relaxed as aR C a*R.Finally,any EP element is characterized in terms of the n-EP property,which is a generalization of the bi-EP property.In chapter 6,several characterizations and properties of core partial order,star par-tial order and diamond partial order in R are given.We show that the pseudo upper semilattice of two Moore-Penrose invertible elements exists under some conditions.Also,the expression of such pseudo upper semilattice is obtained.This is an improvement of the work of R.E.Hartwig in Proc.Amer.Math.Soc.Moreover,the relationships among the core partial order and left star partial order,the diamond partial order and left star partial order are obtained.