In order to study the Drazin invertibility of a matrix with the generalized factorization over an arbitrary ring, the necessary and sufficient conditions for the existence of the Drazin inverse of a matrix are given by the properties of the generalized factorization. Let T = PAQ be a square matrix with the generalized factorization, then T has Drazin index k if and only if k is the smallest natural number such that Ak is regular and Uk（Vk） is invertible if and only if k is the smallest natural number such that Ak is regular and Uk（Vk） is invertible if and only if k is the smallest natural number such that Ak is regular and Uk（Vk） is invertible. The formulae to compute the Drazin inverse are also obtained. These results generalize recent results obtained for the Drazin inverse of a matrix with a universal factorization.
Let φ be a pre-additive category. Assume that φ： X→X is a morphism of φ. In this paper, we give the necessary and sufficient conditions for φ to have the Drazin inverse by using the von Neumann regular inverse for the φ^k, and extend a result by Puystjens and Hartwig from the group inverse to Drazin inverse.
A ring R is called right zip provided that if the annihilator τR（X） of a subset X of R is zero, then τR（Y） = 0 for some finite subset Y C X. Such rings have been studied in literature. For a right R-module M, we introduce the notion of a zip module, which is a generalization of the right zip ring. A number of properties of this sort of modules are established, and the equivalent conditions of the right zip ring R are given. Moreover, the zip properties of matrices and polynomials over a module M are studied.
A ring R is said to be satisfying P-stable range provided that whenever aR ＋ bR = R, there exists y ∈ P（R） such that a ＋ by is a unit of R, where P（R） is the subset of R which satisfies the property that up, pu∈ P（R） for every unit u of R and p ∈P（R）. By studying this ring, some known results of rings satisfying unit-1 stable range, （ S, 2） -stable range, weakly unit 1- stable range and stable range one are unified. An element of a ring is said to be UR if it is the sum of a unit and a regular dement and a ring is said to be satisfying UR-stable range if R has P-stable range and P（R） is the set of all UR-elements of R, Some properties of this ring are studied and it is proven that if R satisfies UR-stahle range then so does any n × n matrix ring over R.
本文主要给出F-复盖的直积还是一个F-复盖的充分条件和充要条件.假设右R-模类F在直积,直和项下封闭,{M_i}i∈I是一簇右R-模.如果每个φ_i:F_i→M_i都是M_i的具有唯一映射性质的F-复盖,且multiply from i∈I M_i有F-复盖,则可以得到是multiply from i∈I M_i的F-复盖.另外我们证明如果φ_i:F_i→M_i是M_i的F-复盖,且multiply from i∈I M_i有F-复盖,则是multiply from i∈I M_iF-复盖当且仅当multiply from i∈I Kerφ~i不包含multiply from i∈I F_i中的非零直和项.从而改进、推广了文[6]中的相应结果.
