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.
