中文摘要：

本文主要在一般线性空间框架中从纯代数的角度研究代数广义逆的可加性与表示问题首先在线性空间中利用空间代数直和分解给出I＋AT＋可逆的充要条件，进而T-＋=T＋（I＋AT＋）-1，给出了T＋具有最简表示的一系列充要条件．其次讨论了在Banach空间广义逆和Hilbert空间Moore-Penrose逆扰动问题研究中的应用．本文的主要结果推广和改进了相关文献中的一些近期成果．

英文摘要：

In this paper, the authors study the additivity and expression of algebraic generalized inverses from the view of pure algebra in the framework of linear space. Utilizing the algebraic direct sum decomposition of linear space, we first give the necessary and sufficient condition of the invertibility of I ＋ AT＋ and T＋ = T＋（I ＋ AT＋）-1. We also provide some necessary and sufficient conditions for T＋ to have the simplest expression. As applications, we discuss the perturbation problem of generalized inverse in Banach space and Moore-Penrose inverse in Hilbert space, which extend and improve many recent results in this topic.