In this paper, we present a method how to get the expression for the group inverse of 2×2 block matrix and get the explicit expressions of the block matrix (A C B D) under some conditions.
In order to study the deformation of algebras the notions of Hom-algebras are introduced.The Hom-algebra is a generalization of the classical associative algebra.First the Hom-type generalization of dimodules which is called the Hom-dimodule is introduced and its properties are discussed Moreover the category of Hom-dimodules in connection with the Hom D-equation R12 R23 =R23 R12 for R∈Endk M⊙M and a Hom-module M is investigated.Some solutions of the Hom D-equation from Hom-dimodules over Hom-bialgebras are given and the FRT-type theorem is constructed in the category of Hom-dimodules. The results generalize and improve the FRT-type theorem in the category of dimodules.
An element a of a ring R is called Drazin invertible if there exists b∈R such that ab =ba,bab =b,and a -a2 b is nilpotent.The element b above is unique if it exists and is denoted as aD .The equivalent conditions of the Drazin inverse involving idempotents in R are established.As applications, some formulae for the Drazin inverse of the difference and the product of idempotents in a ring are given.Hence,a number of results of bounded linear operators in Banach spaces are extended to the ring case.
设R为环,R的右理想I称为小理想如果对任意R的真右理想K都有I+K≠R.环R称为右小内射环如果每个从R的小右理想I到R R的同态可扩张为从R R到R R的同态.左小内射环定义类似.讨论了环的扩张如平凡扩张、形式三角矩阵环、上三角矩阵环等的小内射性.证明了环R通过双模R V R的平凡扩张S=R∝V为右自内射环当且仅当S为右小内射环当且仅当V作为右R-模为自内射模且R=End V R.并证明了非平凡的上三角矩阵环一定不是右小内射环.
Let a, b be two generalized Drazin invertible elements in a Banach algebra. An explicit expression for the generalized Drazin inverse of the sum a + b in terms of a,b,a^d,b^d is given. The generalized Drazin inverse for the sum of two elements in a Banach algebra is studied by means of the system of idempotents. It is first proved that a + b∈A^(qnil) under the condition that a,b∈A^(qnil),aba = 0 and ab^2= 0 and then the explicit expressions for the generalized Drazin inverse of the sum a + b under some newconditions are given. Also, some known results are extended.
Let R be an associative ring with unity 1. The existence of the Moore-Penrose inverses of block matrices overR is investigated and the sufficient ad necessary conditions for such existence are obtained. Furthermore, the representation of the Moore-Penrose inverse of M=[0 A C B]is given under the condition of EBF - 0, where E - I - CCT and F - I -AfA. This result generalizes the representation of the Moore-Penrose inverse of the companion matrix M =[0 a In b]due to Pedro Patricio. As for applications, some examples are given to illustrate the obtained results.
The properties and some equivalent characterizations of equal projection（ EP）, normal and Hermitian elements in a ring are studied by the generalized inverse theory. Some equivalent conditions that an element is EP under the existence of core inverses are proposed. Let a∈R , then a is EP if and only if aa a^# = a^#aa . At the same time, the equivalent characterizations of a regular element to be EP are discussed.Let a∈R, then there exist b∈R such that a = aba and a is EP if and only if a∈R , a = a ba. Similarly, some equivalent conditions that an element is normal under the existence of core inverses are proposed. Let a∈R , then a is normal if and only if a^＊a = a a^＊. Also, some equivalent conditions of normal and Hermitian elements in rings with involution involving powers of their group and Moore-Penrose inverses are presented. Let a∈R ∩R^#, n∈N, then a is normal if and only if a^＊ a^＋（ a^#） n = a^# a＊（ a^＋） ^n. The results generalize the conclusions of Mosiet al.
